Many topological notions (continuity, connectedness, etc.) were
first introduced in the context of a metric space,
where a distance d
is defined which is endowed with the following axiomatic properties:
d(x,y) is a nonnegative real number
(called distance from x to y).
d(x,y) = d(y,x).
d(x,z) never exceeds d(x,y) + d(y,z)
(the triangular inequality ).
d(x,x) = 0
d(x,y) is zero only if x = y
(or else, d is called a semidistance).
From a modern viewpoint, topological properties are based on the concept of an
open set, as discussed in the next article.
In a metric space, an open set is a set which contains an open ball centered
on every point in it.
The open ball of center C and (positive)
radius R is the set
of all points whose distance to C is less than R.
(2007-11-02) Topological Spaces
(Brouwer,
1913)
Defining a topology is singling out some subsets as open.
A set E is said to be a topological
space when it possesses a specific topology.
Formally, a topology is simply a particular collection
of subsets,
called open sets verifying the following axiomatic properties
(L.E.J. Brouwer, 1913).
The empty set (Æ) is open.
The whole set (E) is open.
Any union of open sets is open.
Any intersection of finitely many open sets is open.
In particular, it can be checked that those axioms are verified for the
open sets defined as above in the special case of
metric spaces.
The trivial (or indiscrete or chaotic)
topology is { Æ, E } ;
only Æ
and E are open
(in French, this is called topologie grossière ).
At the other extreme, with the
discrete topology, every set is open.
Neighborhood :
A neighborhood of X (X may be either
a point or a set) is a set which contains some open set containing X.
Open neighborhood :
A neighborhood which is an open set...
Thus, an open neighborhood of X is just an open set containing X.
Open neighborhoods are the only neighborhoods some authors consider.
There are good historical reasons for that viewpoint, but the modern nomenclature
has freed itself from that constraint, so we may speak freely about interesting
things like closed neighborhoods or compact neighborhoods...
Interior :
The interior Å of a set A is the union of all open sets
it contains.
(That's the largest open set contained in A.)
Limit-point :
A point a of a topological space E
is said to be a limit-point of a subset A when
every [open] neighborhood of a intersects A in at least one
point besides A. This concept was introduced by Cantor
in 1872
(in the special case of the metric space of real numbers).
Derived set of a set E :
It's the set E' of the limit-points of E.
(2014-11-26) Limits in a Topological Space
Generalization of the metric notion of a limit.
An element a of a topological space is said to be the limit
of a sequence an
of other points when, for any given integer N, there is a neighborhood
of a which contains every an for
n ≥ N.
A sequence is said to be convergent if it has a limit.
In some coarse topological spaces,
a convergent sequence may have more than one limit.
However, such spaces are infrequently considered in practice
(except, possibly, as a source of counterexamples for honing the definitions of general
topological concepts). Usually, we only bother with Hausdorff spaces,
where the limit of a sequence, if it exists, is necessarily unique.
(2021-08-29) A net is a generalized sequence (French: filet).
A net is a map from a directed set
(usually, to a topological space).
This concept of a generalized sequence was introduced in 1922 by
E. H. Moore (1862-1932) and
Herman L. Smith (1892-1950).
It used to be called a Moore-Smith sequence.
The current name of net was coined in 1955 by
John L. Kelley (1916-1999)
in his graduate textbook General Topology.
A key motivation for introducing nets is that the counterparts with nets
of some specialiazed topological concepts depending on sequences
(sequential continuity, sequential compactness, etc.) match the
general concepts if nets are substituted for sequences
(continuity, compactness, etc.).
(2013-01-04) Basis of a Topological Space
The open sets are all the unions of sets from a topological base.
There's at least one such basis: the topology itself
(the set of all open sets).
The smallest possible cardinality of a basis is
the weight of the topology.
Subbasis :
A set B of open sets is a
subbasis of the topology
if no lesser topology exists where all the elements of B are open sets.
Equivalently, B is a subbasis of the topology if and only if a basis of the topology
is formed by the empty set and all finite intersections of sets of B
(including the nullary
intersection, which is equal to the entire space).
Local Basis :
A local basis
(or neighborhood basis ) at point x is a family of open sets
of which every neighborhood of x contains a member.
A first-countable space is a topological space
for which there's a countablelocal basis at every point.
Every metric space is first-countable
(the open balls of rational radius centered on x form a countable local basis at point x).
Second-Countable Spaces :
A second-countable space is a topological space
for which there's a countable basis.
A second-countable space is clearly first-countable.
A second-countable space is separable.
It's also Lindelöf
(every open cover contains a countable subcover).
In the particular case of metric spaces, the three properties
(second-countability, separability and Lindelöf) are equivalent.
A countable product of second-countable spaces is second-countable.
However an uncountable such product may not even be first-countable.
(2007-11-02) Closed Sets
A subset is closed when its complement is open
In a topological space (as defined above)
a set is said to be closed
when its complement is open
(the complement of a subset F of a set E is the subset of E consisting of all
the elements of E which are not in F).
Clearly, a topology on a set E could be specified
by indicating which of its subsets are closed.
If such a viewpoint is adopted, the closed sets
must simply verify the following axiomatic properties:
The whole set (E) is closed.
The empty set (Æ) is closed.
Any intersection of closed sets is closed.
Any union of finitely many closed sets is closed.
The equivalence of those properties with the axiomatic properties
previously stated for open sets is based on the fact that the
complement of an intersection is the union of the complements,
whereas the complement of a union is the intersection of the complements
(de Morgan's Laws).
One topology which is best defined this way is the so-called
cofinite topology, for which the only closed sets are
Æ, E and all its finite subsets.
Another example is the Zariski topopogy in
algebraic geometry.
Closure :
The closure (French: adhérence)
A of a set A is the intersection of all closed sets
which contain it.
(It's the smallest closed set containing A.)
Border (or boundary) :
The border (or boundary)
¶A of a set A
is the intersection of the closure of A and the complement
of the interior of A.
( ¶A is a closed set,
because it's the intersection of two closed sets.)
Punctured space :
The complement of a single point in a topological space
(such a space can be said to be punctured at that point, if needed).
By extension, we may also consider a space punctured at several discrete points
(it's the complement of those points).
A punctured space can be very different from the whole space topologically
(e.g., punctured torus).
Dense subset :
A set is said to be dense in a topological space when
its closure is equal to the entire space.
Nowhere-dense set :
A set is said to be nowhere dense when
its closure
has empty interior.
Almost-open subset :
A subset is said to be almost open if it has the
property of Baire
(i.e., its symmetrical difference with some open set is a
meager set).
Separability :
A topological space
is said to be separable
if it's the closure of a countable subset
(equivalently, if a countable set is dense in it).
For example, the real line
is separable because it's the closure of the set of rational numbers
, which is
indeed countable.
(2007-12-06) Subspace F of a topological space E.
Open sets of F are intersections with F of open sets of E.
A subspace F of a topological space E
is a subset of E endowed with the so-called induced
topology where an open set of F is defined to be the intersection
with F of an open set of E.
Equivalently, a closed set of F is defined
to be the intersection with F of a closed set of E.
A subspace F is always both open and closed in itself,
but it need not be either open or closed in the whole space E.
(F could be any subset of E).
(2007-11-09) Optional separation axioms
( Trennungaxiome ).
The fewer the open sets, the coarser the topology.
The basic structure of a topological space is not sufficient
to support some general statements
that require various assumptions about how a
topology distinguishes between points.
Historically, some of the following "separation axioms" were once
considered for inclusion in the general definition of a
topological space.
Mercifully, none of them have been retained in that general capacity.
Each of them just denotes a particular class of topologies which is specified
whenever the corresponding properties are needed.
The standard characterizations of all separation axioms are of the form:
Two things separated in a weak sense are separated in a stronger sense.
A subscripted T denotes a choice of a separation axiom
within the most commonly used hierarchy, tabulated below, where
Ti implies Tj if i > j.
The letter T is for Trennung,
a German word meaning "separation".
(Separability is
a different concept, in English and French, at least.)
Most commonly, a topological space is said to be "separated" when is
verifies the "Hausdorff condition"
(i.e., the separation axiom T2 ).
Such a separated topological space is best called a
Hausdorff space.
In a Hausdorff space, a convergent sequence has a unique limit.
Kolmogorov classification, from coarser to finer
(T = Trennung axiom)
The trivialindiscrete or chaotic topology
(where every nonempty open set contains everything) is clearly the
coarsest possible topology.
It doesn't verify any of the above separation axioms
(except if the entire space is empty or contain just a single point).
At the other extreme, the
discrete topology (where every subset is open)
satisfies every conceivable separation axiom. Neither of those two extreme topologies
is very useful, except to provide didactic examples or counterexamples.
On any infinite space,
the cofinite topology
(where the closedproper subsets are just the finite proper subsets)
verifies T1 but nothing stronger.
(HINT: The intersection of two nonempty open sets is infinite.)
Any metric spacedoes satisfy
all of the above separation axioms.
Proofs (or exposed tautologies) :
T1 => T0 Trivially.
T1 <=> { All finite sets are closed. } :
Let F be a finite subset of a T1 space E.
Consider a given point y of F.
For any other point x of E, there's an open set containing x but not y.
The union of all such open sets is an open set containing every point of E except y;
it's thus the complement of {y}.
Having an open complement, {y} is closed. So is F, as a finite union of closed sets.
Conversely, assume that every finite subset of E is closed.
Then, for any ordered pair of distinct points (x,y), the complement of {y}
is an open set containing x but not y. Therefore, E obeys T1
T2 => T1 :
If distinct points x and y have disjoint neighborhoods, we may pick an open
set in such a neighborhood of y which doesn't contain x because it doesn't
intersect the relevant neighborhood of x.
T2 <=> { A pair of distinct points is always a disconnected set. } :
Consider two distinct points x and y in a space obeying T2 .
As they have disjoint neighborhoods, we have two disjoint open sets such that one contains x but
not y and the other contains x but not y.
Thus, {x} and {y} are disconnected from each other.
Conversely, assume that two distinct points x and y are always disconnected from each
other. This means that we have two disjoint open sets which contain x and y respectively.
As those are disjoint neighborhood of x and y,
T2 holds.
(2007-11-09) Compactness (Alexandrov & Urysohn, 1923)
A set is compact when any open cover includes a
finite subcover.
(An open cover is a union of open sets containing the prescribed set.)
A set of compact closure is called
precompact or relatively compact.
Note that a closed subset of a compact set is compact.
Also, the intersection of a closed set and a compact set is compact.
In metric spaces and Hausdorff spaces,
all compact sets are closed,
but this need not be so in general topology.
A counterexample is the set of integers
under the
cofinite topology (where
the nonempty open sets are just complements of finite sets).
With that topology, every subset of
is compact
(HINT: Only finitely many elements are missing from
the first open set in the cover, so only finitely other open sets from the cover
are needed for the rest). This makes any infinite subset compact but not closed.
Historical Origins for Metric Spaces (Fréchet, 1906)
Two closely-related notions are equivalent to compactness
in the case of metric or metrizable spaces:
Sequential compactness :
A set is sequentially compact when any sequence in it contains a convergent subsequence.
Limit-point compactness :
A set is limit-point compact when it contains a limit-point of every infinite subset.
In the Euclidean space n, two famous theorems are equivalent:
Bolzano-Weierstrass theorem
(Bolzano, 1817):
Every bounded sequence of points has a convergent subsequence.
Heine-Borel theorem :
A set is compact iff it's
closed and bounded.
In this, a compact set is defined in the modern sense as "a set
for which any open cover contains a finite subcover" which is
known as the Heine-Borel criterion
(as opposed to the aforementioned Bolzano-Weierstrass criterion defining
sequential compactness, which may not be equivalent to compactness for non-metrizable
topological spaces).
A subset of a metric space is called totally bounded
when it can be covered by finitely
many balls of radius r, for any given radius r.
In a Euclidean space of infinitely many dimensions, a bounded set
(like a ball of unit radius) need not be totally bounded.
Actually, a closed ball is compact only in a space of finitely many dimensions.
A metric space is compact if and only if it's
complete and totally bounded.
Proof (by ChatGPT 3.5) broken down into 3 lemmas:
1/ A compact metric space X is complete :
If X wasn't complete, there would be a Cauchy sequence {x_n} in X that does not converge to any point in X.
Consider an open cover of X consisting of open balls centered at the elements of that Cauchy sequence.
It doesn't include a finite subcover because the sequence does not converge to any point.
2/ A compact metric space X is totally bounded :
Assume X is not totally bounded. Then, there exists some r > 0 for which no finite collection of r-balls can cover X.
You can construct an open cover {U_n} by considering the r/2-balls centered at the points of X.
This open cover has no finite subcover because if it did, you could take the r-ball centered at each point in the finite subcover,
and it would cover X, contradicting the assumption.
3/ A complete totally-bounded metric space X is compact :
To prove this, consider an open cover {U_i} of X. We want to show that there exists a finite subcover.
Since X is totally bounded, there exists a finite subset {x_1, x_2, ..., x_n} of X
such that the 1-balls centered at these points cover X.
These balls can be denoted as B(x_1, 1), B(x_2, 1), ..., B(x_n, 1).
Now, consider each of these balls.
Since X is complete, every Cauchy sequence in X converges to a point in X.
For each ball B(x_i, 1),
construct a Cauchy sequence by selecting points from the open cover {U_i} that intersect this ball.
Since these balls cover X, each point in X will belong to at least one such ball.
By completeness, each of these Cauchy sequences converges to a point in X.
Now, take a finite subcover {U_{i_1}, U_{i_2}, ..., U_{i_k}} from the open cover {U_i}
such that each of these chosen open sets corresponds to one of the balls B(x_j, 1) where 1 ≤ j ≤ n.
Since each Cauchy sequence converges to a point in X, you can choose, for each i,
a point from the Cauchy sequence corresponding to B(x_j, 1) that lies within U_{i}.
This gives you a finite subcover.
(2020-10-03) Paracompact Spaces (Dieudonné, 1944)
For such a space, every open cover has a locally finite open refinement.
Some authors also require a paracompact space to be Hausdorff.
Dieudonné proved that every paracompact Hausdorff space is a normal Hausdorff space (T4 ).
D is said to be a refinement of a cover C when every set of D
is contained in a set of C (it need not be equal to it).
A cover is said to be
locally finite
when every covered point possesses
a neighborhood which intersects only finitely many
sets from the cover.
A compact set is necessarily paracompact
(because a subcover is a refinement and a finite subcover is locally finite).
The converse isn't true.
Every closed subspace of a paracompact space is paracompact.
The product of paracompact spaces need not be paracompact,
but the product of a compact space and a paracompact space is paracompact.
In a metric space, all subsets of a paracompact set are paracompact.
Nagata-Smirnov Metrization Theorem :
A topological space is said to be metrizable
when it's homeomorphic to a metric space.
A topological space is said to be locally metrizable
when every point had a metrizable open neighborhood.
A locally metrizable space is metrizable iff
it's paracompact and Hausdorff.
Corollary :
A manifold is metrizable if and only if it is paracompact.
(2007-11-17) Locally Compact Spaces
Spaces in which every point has a compact neighborhood.
As is demonstrated by the Heine-Borel Theorem
for metric spaces, compactness and completeness are strongly related but compactness
implies an overall limitation which is not present in the purely local
concept of completeness.
Traditionally, completeness is only defined for metric spaces
(because Cauchy sequences are a purely metrical concept).
A loose counterpart of completeness in general topological spaces,
must involve some concept of local compactness.
All the definitions which have ever been proposed are equivalent
to the one featured above in the case of Haudorff spaces.
Again, local compactness is a relatively minor topological concept
which is only loosely related
to the very important metric concept of
completeness
(which André Weil
extended to uniform spaces in 1937).
Topological Vector Space :
A topological vector space
is locally compact iff it's finite-dimensional.
This classical result is due to Frédéric Riesz
(Riesz Frigyes, in Hungarian).
(2012-11-13) The extreme-value theorem
(Bolzano, c.1830)
A continuous real-valued function defined on a compact set
is bounded and attains both of its extreme values.
The original theorem (for functions defined in n-dimensional Euclidean space) was proved in the 1830s
by the Germanophone Bohemian mathematician Bernard Bolzano
(1781-1848) whose relevant work (Functionenlehre) was only published in
1930.
That theorem was established independently by
Karl Weierstrass (1815-1897) in 1860.
Bolzano's proof consisted of two steps: First, he showed that a continuous function
over a compact domain had to be bounded. Second,
he proved that the least upper bound and greatest lower bound are attained at some points.
For both steps, Bolzano invoked what's now called the
Bolzano-Weierstrass theorem
("every bounded sequence contains a convergence subsequence") which he had established in 1817,
as a lemma in the proof of his version of the
intermediate value theorem
("a continous function with negative and positive values must vanish at some point").
Nowadays, to establish the theorem for compact sets in any topological space
(not necessarily a metric or metrizable one) we merely observe
that the continuous image of a compact set of points
is a compact set of reals, namely a closed bounded set of real numbers.
Thus, there is a minimum and a maximum and both are the images of some points...
A continuous image of a compact set is compact.
Proof : If f is continuous and
A is compact, consider any open cover of
f (A). The preimages of the
elements of that cover are open sets (because f is continuous)
which cover A. Since A is compact,
we can select a finite number of those which cover A.
The images of that selection are open by construction and form a finite subcover of the original cover
of f (A).
The converse isn't true: Functions that send compacts to compacts aren't necessarily
continuous. For example, any function that takes on only finitely many values has
this property (since finite sets are compact) but is not necessarily continuous.
(2012-09-21) Borel Sets. Borelian Tribe. (1898)
The two definitions of a Borel set are usually equivalent.
A s-algebra or tribe
(the French term
tribu
was introduced in 1936 by the BourbakistRené de Possel, 1905-1974)
is a family of sets closed under countable intersection,
countable union and relative complement.
The open sets need not form a tribe.
The smallest tribe containing all open sets
(the intersection of all tribes that contain all open sets) is
called the Borel tribe or the Borelian tribe.
Its elements are called Borel sets or Borelians.
Borel sets are thus derived from open sets by countable union, countable
intersection and relative complement.
Equivalently, we may start with the closed sets. The same Borel tribe
is also obtained as the smallest tribe containing all closed sets.
Some authors have proposed to define the Borel tribe
as the smallest tribe containing all the compact sets.
This definition is usually equivalent to the classical
definition presented above, but for some pathological
topologies, this ain't so...
U is closed if and only if
it contains the limit of all convergent sequences of its own points.
U is compact when any sequence of its points has a subsequence
which converges in U.
U is completeiff
any Cauchy sequence of points of U converges in U.
(2007-11-02) Continuous Functions
A function is continuous iff the
inverse image of any open set is open.
Arguably, one of the original motivations of the entire field
of topology was to characterize continuity in very general terms.
This is achieved by the above definition, which looks natural only after
years of proper mathematical training...
Two equivalent statements characterize a
continuous functionf defined over some subset D
of one topological space with values in another:
The inverse image of any open set is a set which is open in D.
The inverse image of any closed set is a set which is closed in D.
Recall that a set "open (resp. closed) in D"
is the intersection with D of an open set (resp. a closed set).
Continuous functions verify two important properties, respectively known as
the extreme value theorem and the
intermediate value theorem
(at least, that's the name they have for real functions of a real variable). Namely:
However, neither statement is characteristic of continuous functions
(i.e., each can be satisfied by some discontinuous functions as well).
(2007-10-31) Homeomorphisms bicontinuous functions Those are continuousbijections whose inverses are continous too.
A homeomorphism is a bicontinuous
function (which is to say that it's continuous and
bijective and that its inverse is continuous as well).
An homeomorphism can be construed as an isomorphism
of the topological structure. A bijection is an homeomorphism if and
only if it transforms any open set into an open set and any closed set
into a closed set.
Two topological sets are said to be homeomorphic
when there's an homeomorphism between them. Two homeomorphic spaces
have identical topological properties.
(2020-01-02) Hilbert Curve
Continuous mapping from the segment ]0,1[
to the open square ]0,1[ 2
Here, a curve is understood to be
a continous function from the open interval
into anything. Until Peano
proved otherwise in 1890 by describing the first
space-filling curve,
it was thought that the image (or range) could only be a manifold of dimension 1.
The next year (1891) Hilbert improved the aesthetics of Peano's result with a very symmetrical curve
filling the entire (open) unit square. This is now known as the Hilbert curve.
The Hilbert curve is an example of a continuous bijection
whose inverse is not continous (at any point).
A continuous bijection whose inverse is also continous is called bicontinuous.
A bicontinuous function is also called an homeomorsphism.
All topological properties are (by definition) preserved under homeomorphisms.
Dimensionality is one of them.
A one-dimensional segment isn't homeomorphic to a two-dimensional square,
in spite of the existence of the Hilbert curve.
(2012-12-27) Restricting or extending a
continuous function.
A continuous function restricted to a subspace remains continuous.
The restriction of a continuous function to a
topological subspace
is always continuous. However, there may not always
be a way to extend a continuous function defined on a subspace
to a continous function defined on the whole space...
One example (butchered
by educators with weak topological skills)
pertains to the expression f (x) = 1/x
(a simple homographic transformation).
This relation does define a
continuousbijectionf from D to D
when D is one of the following sets.
(Continuity is counterintuitive in the first case!)
U
{¥}
(reals with single unsigned infinity; topology of a circle).
*
(the nonzero complex numbers).
U
{¥}
(projective complex line; topology of a sphere).
However, no continous extension of f can be defined from
(all real numbers, including zero)
to the closed interval
[-¥, +¥] endowed with the usual
topology (where the positive reals do not constitute
a neighborhood of negative infinity, and vice-versa).
This is why three different types
of pseudo-numerical infinities are defined (or ought to be defined)
in computer algebra systems (CAS):
¥ = 1/0 :
Unsigned, complex or algebraic infinity.
+¥ and -¥
: Signed infinities (used mostly in realanalysis).
(2007-11-09) Product Topology,
Tychonoff Topology (1926, 1935)
Coarsest topology for which all projections are continuous.
Consider the cartesian product E
of finitely or infinitely many sets:
E =
Õ
Ei
iÎI
For each index i, there's a projection function
pi which transforms an element x of E
into the corresponding component of x in
Ei . Formally:
{ x } =
Õ
{ pi (x) }
iÎI
E is best endowed with the least topology
which makes all such projections continuous.
(Recall that the "topology" is, formally, the collection of
all open sets.)
This so-called product topology
can also be described as consisting of all unions (finite or infinite)
of finite intersections of sets of the following form:
Õ
Ui
where Ui is an open subset of Ei
which is different from Ei in only
finitely many cases.
iÎI
If we didn't insist on Ui being a
proper open set for only
finitely many indices, we would obtain a finer
topology known as the
box topology.
The above product topology is often called the Tychonoff topology
(it's the initial topology
with respect to the projection maps).
It was discovered by Andrei Nikolaevich Tikhonov
(1906-1993) in 1926, before he even graduated...
Arguably, this is the only "correct" topology to consider over a cartesian product of topological spaces.
In particular, it ensures that a map f is continuous if and only if
its components fi are continuous.
{ f (x) } =
Õ
{ fi (x) }
iÎI
This desirable theorem would not be true, in general, with the box topology,
which is too fine
and makes it much harder for a function to be continuous.
Similarly, Tikhonov proved that his product topology makes
any product of compact spaces compact.
By comparison, box topology looks like a misguided idea
(except for a finite cartesian product, or when almost all
components are endowed with the trivial topology, in which
cases the two concepts coincide).
The cartesian product of any collection of compact spaces is compact.
This is one of the most important results of general topology.
It helped define the modern concept of compactness based on the
Heine-Borel criterion (every open cover has a finite subcover).
That definition replaced a definition of compactness, now called
sequential compactness, based on the
Bolzano-Weierstrass criterion (any sequence has a convergent
subsequence).
Both definitions are equivalent for metrizable spaces
but neither implies the other for [some?] other topological spaces.
For example, the product of an uncountable number of copies of the closed unit interval
fails to be sequentially compact.
Tychonoff 's theorem relies on the
Axiom of choice.
In fact, Tychonoff 's theorem and the Axiom of choice
turn out to be equivalent statements.
Cauchy's Mistake (1821) :
Cauchy
thought that a function of two variables x and y
which is continuous with respect to x and with respect to y
must be continuous with respect to (x,y).
This ain't so. The following counterexample was produced in 1870
by Johannes Thomae (1840-1921).
It has continuous projections but is discontinuous at the (0,0) point.
(HINT: On the line y = a x,
the function f has a constant value which depends on a.)
(2007-11-02) Connected Set
A connected set cannot be split by two disjoint open sets.
By definition :
Two sets are said to be disconnected from each other
if they are respectively contained in two disjoint open sets.
A set is said to be disconnected
if it's the union of two nonempty
parts that are disconnected from each other.
A set is said to be connected if it's not disconnected.
To prove that a set A is connected, we may show
that it can't be contained in the union of of two disjoint open sets U
and V unless one is empty.
A nonempty topological space E
is connected if and only if it doesn't contain any
clopen (i.e., both open and closed)
nonempty proper subset.
The empty set is connected. So is any set containing only one point.
A topological space where there are no other connected sets is said to
be totally disconnected (in such spaces,
there are nonconnected sets with more than one point).
For example, the discrete topology always
produces a totally disconnected topological space.
More interestingly, the followings spaces are totally disconnected:
On the other hand, with the trivial topology
(the so-called indiscrete topology) every set is connected.
The closure of a connected set is connected.
Proof : If the closure of a set is disconnected,
then that closure can be split by two disjoint open sets which
also split the set, proving it's disconnected.
Connected Components :
A connected component of a topological space is
a maximal connected nonempty set
(i.e., a nonempty connected set which isn't
contained in any larger connected set).
The convention is thus made that the empty set
isn't a connected component of anything
(not even itself) although it's definitely connected.
All the connected components form a unique partition
of the topological space (i.e., they're pairwise disjoint and
their union is the whole space).
This fundamental property is the reason why we had to rule out
the empty set as a connected component
(since elements of a partition are never empty).
An empty collection of sets is a valid partition which
corresponds to the connected components of an empty topological space
(which has no connected components, as previously noted).
Every connected component must be closed
(HINT: its closure is connected).
If there are only finitely many of them, each is also open
(HINT: its complement is a finite union of closed sets).
If there are infinitely many connected components,
they're not necessarily open (e.g., Cantor set).
Connectedness and Continuity :
The
above definition of continuity satisfies the intuitive
requirement that a continuous function must transform a connected set into a connected set...
A continuous image of a connected set is connected.
Proof :
Let f be a continuous function. By definition,
it is such that the inverse image
of any open set is an open set.
We have to prove
that the direct imagef (A)
of a connected set A is connected or, equivalently,
the contrapositive statement:
If f (A) is disconnected, so is A.
Well, if f (A) is disconnected, we can split it into two
nonempty parts respectively contained in two disjoint open sets U and V.
Consider the open sets f -1 ( U )
and f -1 ( V )...
Neither has an empty intersection with A, because U
and V both have at least one element from f (A).
Every element of A is in one or the other of those two open sets.
Their intersection is empty,
because any element of both would need to have its image in both U
and V, which are disjoint.
A is thus split into nonempty parts by two disjoint open sets.
Note, however, that the converse is false:
There are discontinuous functions which transform every
connected set into a connected set. The
following section provides many such examples in the
special case of real functions of a real variable...
(2014-09-01) Intermediate-Value Theorem
A continuous function from reals to reals maps an interval to an interval.
On the real line, the connected sets are the
intervals. So, that's just
a special case of the general result established in the
previous section
(a continuous image of a connected set is connected).
By contrast, the intervals of rationals are not connected
(the set of rational numbers is totally disconnected)
and there is no equivalent of the intermediate-value theorem for rationals.
Spelled out in elementary terms, this yields a very useful result which
says that, for any continuous function of a real variable
defined between a and b,
any value y between f (a) and f (b)
is equal to f (x), for some x.
A popular formulation used by Bolzano (for functions on an interval) is:
Continuous functions with positive and negative values vanish somewhere.
The converse isn't true :
The intermediate-value property is not
a characteristic property of continuous functions:
There are functions which are not continuous for which the property holds.
Such is the case for any discontinuous derivative f '
of a differentiable function. For example, we may use:
f (x) = x 2 sin ( 1/x )
[ with f (0) = 0 ]
Proof : Since any differentiable function f is continuous,
the extreme value theorem states that it must reach
a minimum and a maximum within any interval [a,b]
on which it is defined. If f ' (a) and
f ' (b) have opposite (nonzero) signs, at least one of
those extrema is not located at an extremity, so it must be at a point x
where f ' (x) = 0
A subset Y is said to be path-connected
(or pathwise connected )
when such a path exists whose image is contained in Y,
for any pair of extremities {a,b} in Y.
Some authors have used the terms arc-connected
(or arcwise connected ) for that same concept.
However, this practice is not recommended, since the term is best used for
a stronger concept.
A set consisting of a single point is path-connected. The empty set isn't.
(Thus the empty set is a trivial example of a connected set
which isn't path connected.)
A nontrivial example of a connected set which isn't path-connected is
the closure of the so-called
topologist's sine curve ; the
planar curve of cartesian equation:
y = sin ( 1/x ) for 0 < x
< 2/p
That closure includes the segment
at x = 0, between (0,-1) and (0,1).
Lemma :
The interval [0,1] is connected (proof by contradiction).
Theorem :
Any path-connected set is connected.
Proof :
Consider two arbitrary points
a and b
of a path-connected set Y.
Let P be the image of a path joining them within Y.
If the two extremities were respectively in two disjoint open
sets U and V whose union contained Y,
then those two open sets would likewise split P and prove it
to be disconnected. Since we know that P
is connected (as a continuous image of the connected
set [0,1] examined in our lemma)
we deduce that a and b
cannot possibly be in two disjoint open sets covering Y.
As this is true of any pair of points of Y, there cannot be two nonempty
parts of Y in disjoint open sets covering Y.
Therefore, Y is connected.
Lemma :
In a normed space, balls
and convex sets are path-connected.
(HINT:
Consider the pathf (u) = (1-u) a + u b )
Theorem :
In a normed space,
a connected open set is path-connected.
Proof :
For any point a of a nonempty open set U,
we may define the following two sets V and W.
Both are open.
(HINTS:
Open balls are path-connected (lemma).
The union of two arcs sharing an extremity is an arc.)
V consists of all points b
of U for which there is a path
from a to b.
W consists of all points z
of U for which there's no path from
a to z.
U is the union of the two disjoint open sets V and W.
V is nonempty, since it contains a.
Therefore, if U is connected, W must be empty,
which means that there's a path from
a to any other point of U.
(2012-12-30) Arcs & Arc-Connectedness
An arc is a topological subspacehomeomorphic to [0,1].
An arc is said to join its pair of extremities
(defined as the image of {0,1} under any
homeomorphism between the arc and the
interval [0,1] ).
A topological space where any point is joined to any other point by an arc
is said to be arc-connected or arcwise connected.
Clearly, an arc-connected space is path-connected
(since bicontinuous functions are continuous).
However, the converse need not be true.
A simple counterexample is a topological space X consisting
of just two points under the trivial topology:
That space X is not arc-connected
because there are no
injections from [0,1] to X,
because [0,1] has more than two elements
(hence no bicontinuous functions between [0,1] and the only pair of points in X).
On the other hand, there's a continuous path from one point of X
to the other, obtained from a function which is equal to one point of X at
zero and the other point elsewhere.
(That function is continuous because the inverse image of the only nonempty
open set of X is equal to [0,1], which is open in itself.)
(2007-11-05) The Fundamental Group
(first homotopy group ) P1 (X)
The homotopy classes of the loops going through a given base point.
(Same groups for arc-connected base points, up to inner isomorphism.)
(2007-11-06) Homotopy Groups
Generalizing the fundamental group to
n-dimensional hyperloops.
(2007-11-06) Diffeomorphisms
Differentiable maps with differentiable inverses.
(2007-10-31) Homology & Cohomology
(J. T. of Summerville, SC. 2000-11-19)
How many edges (lines) are in a cylinder?
I assume we're talking about a finite cylinder; the "ordinary kind"
with two parallel bases, which are usually circular (as opposed, say,
to an infinite cylinder with an infinite lateral surface and no bases).
The answer is, of course, that there are two edges, the two circles.
I think you figured this out by yourself and did not need anybody to tell you,
so I suppose your real concern is elsewhere...
Because you used the term "edges" I suspect you think you've found an
exception to the Descartes-Euler formula, which states that "in a polyhedron"
the numbers of faces (F), edges (E) and vertices (V)
are related by the formula: F-E+V=2.
In a way, you have such a "counterexample": In a cylinder, there are 3 faces
(top, bottom, lateral), 2 edges (top and bottom circles) and no vertices,
so that F-E+V is 1, not 2! What could be wrong?
Nothing is wrong if things are precisely stated.
Edges and faces are allowed to be curved, but the Descartes-Euler formula
has 3 restrictions, namely:
It only applies to a (polyhedral) surface which is topologically "like" a sphere
(imagine making the polyhedron out of flexible plastic and blowing air into it,
and you'll see what I mean). Your cylinder does qualify (a torus would not).
It only applies if all faces are "like" an open disk.
The top and bottom faces of your cylinder do qualify, but the lateral face
does not.
It only applies if all edges are "like" an open line segment.
Neither of your circular edges qualifies.
There are two ways to fix the situation.
The first one is to introduce new edges and vertices artificially to
meet the above 3 conditions.
For example, put a new vertex on the top edge and on the bottom edge.
This satisfies condition (3),
since a circle minus a point is "like" an open line segment.
The remaining problem is condition (2); the lateral face is not "like"
an open disk (or square, same thing).
To make it so, "cut" it by introducing a regular edge between
your two new vertices.
Now that all 3 conditions are met, what do we have?
3 faces, 3 edges and 2 vertices.
Since 3-3+2 is indeed 2, the Descartes-Euler formula does hold.
The better way to fix the formula does not involve introducing
unnecessary edges or vertices.
It involves the so-called Euler characteristic,
often denoted c (chi):
The Euler Characteristic c
( chi )
The fundamental properties of c (chi)
may be summarized as follows:
Any set with a single element has a c of 1 :
"x,
c ( {x} ) = 1
c is additive:
For two disjoint sets E and F,
c(EÈF)
= c(E) + c(F)
If E is homeomorphic to F, then
c(E) = c(F)
("Homeomorphic" is the precise term for topologically "like".)
Using the above 3 properties as axioms, it's easy to show by induction
that, if it's defined at all, the c of n-dimensional
space can only be (-1)n.
(HINT: A plane divides space into 3 disjoint parts; itself and 2 others...)
c (point) = 1
c (entire straight line, or open segment) = -1
c (plane or open disc) = 1
c (space or open ball) = -1
c (space with n-dimensions) =
(-1)n
c (surface of a sphere) = 2
c (surface of an infinite cylinder) = 0
c (surface of torus) = 0
c (circle, or semi-open segment) = 0
etc.
Now, back to our problem:
Why is the Descartes-Euler formula valid to begin with?
Well, that's because the c of a sphere's surface is 2
and it's "made from" disjoint faces, edges and vertices, each respectively with a
c of 1, -1 and 1.
In the "natural" breakdown of your cylinder (whose c
is indeed 2), you have no vertices, two ordinary faces (whose
c is 1) and one face whose c is 0
(the lateral face), whereas the c of both edges is 0.
The total count does match.
Note
(2000-11-19) :
The orthodox definition of the Euler-Poincaré characteristic does not
use the above 3 fundamental properties as "axioms" but instead is closer to the
historical origins of the concept (generalized polyhedral surfaces).
It would seem natural to extend the definition of c
to as many objects as the axioms would allow.
This question does not seem to have been tackled by anyone yet...
Consider, for example, the union A of all the intervals [2n,2n+1[
from an even integer (included) to the next integer (excluded).
The union of two disjoint sets homeomorphic to A can
be arranged to be either the whole number line or another set homeomorphic to A.
So, if c(A) was defined to be x, we would have:
x = x + x and
-1 = x + x
Thus, x cannot possibly be any ordinary number,
and the latter equation says x is nothing like a signed infinity either [as
(+¥)+(+¥) ¹ -1].
At best, x could be defined as an unsigned infinity
(¥) like the
"infinite circle" at the horizon of the complex plane
(¥+¥ is undetermined).
This could be a hint that a proper extension of c would
have complex values...
(2003-11-27) Generalized Euler Characteristic
Extending the Euler characteristic (1752) to complex values.
Just about 3 years after posting the previous article
at its original location, we resumed our reflection
about an extended Euler characteristic.
The hunch about complex values turned out to be decisive,
based on our previous observation that the c of the set A
described in the footnote could only be an unsigned infinity...
In the original version of the footnote,
we shyly called this a "lame" hint that extended chi-values could be complex.
We've now edited this out!
The set A was clearly a failed attempt at building something with a
c of ½.
[As I recall, finding out it could only be an unsigned infinity was
disappointing...]
With hindsight, it's clear that there's a more compelling approach, based on another
well-known property of c concerning cartesian products,
which is worth preserving in any interesting extension of c:
c ( E ´ F ) =
c(E) c(F)
Using the 3 "axioms" of the previous article [and the value
(-1)n which they impose for the c of ordinary
n-dimensional Euclidean space]
this relation can be easily established by [structural] induction
for all "polyhedral" sets.
(Such sets, which are the usual
domain of definition of c,
consist of finite unions of disjoint components,
each homeomorphic to some n-dimensional Euclidean space,
which are called its vertices, edges, faces, cells...)
Therefore, the above relation does not contradict our three axioms and may be
use as a fourth axiom in a larger scope of more general sets,
which remains to be defined...
As we expect complex numbers to be involved,
we're also expecting an arbitrary choice between i and -i,
probably linked to the chirality of sets so that
the chi of a set and of its "mirror image" are complex
conjugates of each other.
We are thus led to assume that c is only preserved
by homeomorphisms that conserve chirality and could restate the third axiom (C) accordingly,
in terms of those homeomorphism which preserve the
orientation of an immersing space.
For an homeomorphism which does not preserve such an orientation, it may be possible to
find a larger space in which the orientation is preserved whose restriction to a smaller
space violates orientation (a two-dimensional symmetry about a line is
a restriction to the plane of a three-dimensional rotation about that line).
This is a clue that an intrinsically chiral topological space
can't be immersed in a space of finitely many "dimensions".
Let's try to build a set E whose cartesian square
E´E has a c of -1...
We would then expect the cartesian product of E and its mirror image to have a
c of +1 and this may guide the search...
Consider a Hilbert space with the countable basis
denoted |0>, |1>, |2>, |3>, etc. It is homeomorphic to its own cartesian square
(HINT: Use the even coordinates of a given ket to form
a first ket and the odd ones to form a second ket.)
(2007-10-31) The Real Projective Plane &
Boy's Surface (1901)
Werner Boy found a 3D immersion of the real projective plane.
The set whose elements are straight lines going through
the origin in three-dimensional Euclidean space is known
as the real projective plane.
David Hilbert (1862-1943)
did not think the real projective plane
could be immersed
as an ordinary surface in 3-dimensional Euclidean space,
but he couldn't prove it was impossible.
So, he assigned the task to one of his graduate students,
Werner Boy,
who earned his Ph.D. in 1901
by finding such an immersion, now called Boy's surface.
Boy's surface
has Euler characteristic
c = 1.
It can be represented as a single-sided surface with a vertical axis of ternary symmetry.
On that axis is a single pole P which looks like an ordinary point from the top
but appears from the bottom as the triple point T
where three seams meet (each such seam is locally equivalent to three flat surfaces
sharing an edge, two of those can be smoothly aligned to allow a vantage point
where all seams are hidden behind a smooth part of the surface).
Other representations do not break at all any fundamental ternary symmetry.
Werner Boy (1879-1914)
worked several years
as a teacher in a Gymnasium of Krefeld (19 km NW. of Düsseldorf)
before returning to his birth town of Barmen (now Wuppertal, 28 km
E. of Düsseldorf).
He died in the first weeks of World War I (September 6, 1914).
(2017-07-25) Classification Theorem for connected Closed Surfaces A closed surface is a compact 2-D manifold without boundary.
A (topological) surface is simply a two-dimensional manifold.
Unfortunately, the traditional locution closed surface doesn't
necessarily apply to a surface which is merely closed;
it has to be compact and borderless as well
(i.e., its boundary must be empty).
Connected Sum of two Surfaces :
The connected-sum of two surfaces is obtained by removing an open disk from one
and a closed disk from the other and gluing
the two remaining component so that the two deleted disk share the same border.
(2006-08-26) Hadwiger's Theorem (1957)
About the additive continuous functions of d-dimensional rigid bodies.
All the finite unions of convex sets of points
in d-dimensional Euclidean space form what's called
the d-dimensional convex ring
(a set of points is convex if it contains all the
straight segments whose extremities are in it).
A map is said to be conditionally continuous when the images
of the convex approximations to a convex body always converges to the image of the body.
The blog of Dan Piponi may serve as a
nice
introduction to Hadwiger's Theorem,
a celebrated theorem of integral geometry published in 1957 by the Swiss
mathematician Hugo Hadwiger (1908-1981). Namely:
If it is invariant under translations and rotations in d-dimensional euclidean space,
any finitely additive function
which maps finite unions of convex bodies
to real numbers must be a linear combination of d+1
uniquely defined
"n-dimensional content" functions (where the index n goes from 0 to d).
Hadwiger's "0-dimensional content" function is proportional to the
Euler characteristic
(the chi-function c ) discussed
above.
In 3 dimensions, the "n-dimensional contents" for n=1,2,3 are respectively
proportional to the body's mean curvature, its surface or its volume.
(2006-04-24) Oriented Topological Spaces
A topological definition of orientation applies to some spaces.
(2007-11-04) Winding number
Counterclockwise turns of a planar curve around an outside point.
Consider a continuous oriented planar curve
which doesn't go through point O.
g : [ a, b ]
®
2 -{O}
=
*
As the point M = g(t)
moves positively along that curve (which need
not be differentiable) an observer at the origin O may
record unambiguously
the variations of the angle which OM forms with
some fixed direction The usual ambiguity modulo
2p does not apply because we are considering
a continuous
variation in an angular difference which starts at zero.
The total change in that angle, expressed in turns
(the number of radians divided by 2p)
is called the winding number
W ( g, O ) of
g around point O.
If the curve is closed, that winding number is an integer.
For example, it's +1 for any counterclockwise circle going around
the origin. It's -1 if such a circle is oriented clockwise.
It's 0 if the circle does not go around the origin.
The winding number
around the origin is invariant under reparametrization of the curve
and also under
homotopy within the
punctured plane
*.
This is illustrated by the following popular theorem:
If a man and a dog walk respectively around closed curves
g0 and
g1 so that
the "leash" segment
[ g0(t), g1(t) ]
never touches the "hydrant" O, then:
W ( g0 , O )
=
W ( g1 , O )
This is a consequence of the invariance of the winding number
by homotopy, since the following curve is a valid
homotopic interpolation
within the punctured plane
(since g(t) is never on the
"hydrant", because it's a point of the "leash").
g(t,s) =
(1-s) g0(t) +
s g1(t)
In particular, the lemma applies whenever the distance from the hydrant
to the man (or to the dog) is greater than the length
of the leash (such an inequality ensures that the hydrant
cannot be between the man and the dog).
Any complex polynomial
P of degree n > 0 has at least one
complex root.
Proof :WLG, assume that P is a monic
polynomial of leading term xn.
We aim to apply the above dog on a leash lemma.
For some positive number r, let's consider the following closed
curves (as the parameter t goes from 0
to 2p).
g0(t) =
( r e it ) n
g1(t) =
P ( r e it )
The winding number of g0
around the origin is clearly n because the argument
of g0(t)
increases by 2pn
when t increases continuously
by 2p.
Now,
|g1(t)-g0(t)|
is bounded by a fixed polynomial in r of degree n-1.
For a large enough r, that's less than
|g0(t)| = r n.
Therefore, by the dog-leash lemma,
the winding number of g1
around the origin is n
(the same as g0 ).
If P didn't have any zeroes, then
g(t,s) =
P ( r (1-s) e it )
would be a valid homotopic interpolation
(within the punctured complex plane)
shrinking g1
down to a pointlike curve located at P(0).
This would make the winding number of
g1 equal to
0 instead of n.
Therefore, P must have at least one zero.
(2007-11-11) Fixed-Point Theorems
For some sets, all continuous mappings have a fixed point.
A fixed-point of a mapping f is
a point x such that f (x) = x.
An idiomatic way to state the same thing is to say that
f fixes x.
The archetypal fixed-point theorem is
Brouwer's fixed-point theorem, which says that any continuous
function from E to itself must have a fixed point when
E is homeomorphic
to a closed ball of an n-dimensional Euclidean space.
Compact convex (Schauder)
Locally convex (Tychonoff)
In a general topological space,
it may be convenient to turn the fixed-point theorem
into a definition of a specific class of sets.
Let's just call Brouwerian a set E for which all continuous
functions from E to itself have at least one fixed point.
The above fixed-point theorems can be expressed by stating that the following sets are
Brouwerian :
A set homeomorphic to an n-dimensional closed ball. (Brouwer)
A convex compact subset of a normed vector space. (Schauder)
A locally convex subset of a topological vector space. (Tychonoff)
Sperner's lemma (1928):
Combinatorial equivalent of Brouwer's fixed-point theorem.
(2007-11-04) Turning Number
The winding number of
the nonvanishing tangent to a planar curve.
If a point moves in the plane with a continuous nonvanishing velocity,
the winding number of the velocity
about zero-velocity is well-defined
(it would not be for a curve with singular points where the
velocity can vanish, because the winding number is not defined
for a curve which goes through the origin).
This number does not depends on the details of the motion, except
its orientation. It's a characteristic of the oriented
trajectory called the turning number.
For a closed trajectory, that
turning number is an integer.
One way to compute this integer for a closed curve
is to focus only on those points where
the oriented tangent has a given direction (e.g., due east).
Each such point is assigned a zero value if it's an inflection point,
a value +1 if the curve lays to the left of its tangent or
a value of -1 if it lays to its right.
The sum of those values is equal to the curve's
turning number.
The Whitney-Graustein theorem states that two closed
differentiable curves are homotopic within the plane
if and only if they have the same turning number.
(2007-10-26) Eversion of the sphere
A regular homotopy can turn a sphere inside out.
The approach presented in the previous article for closed planar curves
can be adapted to three-dimensional oriented
smooth surfaces by focusing
only on those points where the normal vector is vertical
and pointing upward.
Such points are counted for +1 if the surface does not cross
its tangent plane, whereas proper saddlepoints are counted as -1.
The sum of all such values is a characteristic of the surface which
remains invariant by any homotopy.
A proper
saddlepoint is characterized by a second-order variation which
is a differential form with two real roots.
In 1957,
Steve Smale
(b. 1930, Fields Medal in 1966)
proved that an eversion of the sphere was possible
(this result is so surprising that it's still known as
Smale's paradox).
In 1961, Arnold Shapiro came up with the first practical
eversion of a sphere. He did not publish it but described it to
Bernard Morin. Morin discussed it with René Thom who exchanged
letters about the subject with Tony Phillips.
In 1966, this culminated in a popular article by Philips for
Scientific American
(loosely following Shapiro's original construction).
In 1967, Morin came up with an eversion which was simpler than all previous ones.
Bernard Morin (1931-2018)
has been blind since age 6.
He was a brilliant French mathematician who spent most of his
career at the University of Strasbourg.
(François Apéry,
son of Roger Apéry,
was one of his graduate students.)
He described what's now called the
Morin surface
as the half-way stage in a superb eversion of the sphere.
Morin is also known for having given (in 1978) the first parametrization
of Boy's surface
(the 3D immersion of the real projective plane
found in 1901 by Werner Boy, a student
of David Hilbert).
In 1974, Bill Thurston
(1946-2012) introduced a new sphere eversion
based on his corrugation method (illustrated in the video
Outside in).
(2007-10-31) Classification of Closed Surfaces
& Conway's ZIP (1992)
Conway's "Zero Irrelevancy Proof" of the classification theorem (1860).
A connected closed surface is homeomorphic to either
A sphere with n handles
(orientable, c = 2-2n ). Such a surface is
called an n-torus: sphere (n=0), torus (n=1), double-torus (n=2),
triple-torus, etc.
A sphere with n crosscaps
(nonorientable, c = 2-n ). Such a surface is
called an n-cross surface; a term due to John Conway (1937-2020):
The real projective plane is a cross surface,
the Klein bottle is a double-cross surface,
Dyck's surface is a
triple-cross surface.
Conway's ZIP
by George K. Francis & Jeffrey R. Weeks
(2007-10-31) Braid Group Bn
Emil Artin (1898-1962)
There is a surjective group homomorphism from Bn to
the symmetric group Sn
(the group of all permutations of n elements).
The kernel of this group is the
pure braid group on n strands Pn.
(2020-05-27) Zariski topology
The topology normally used in algebraic geometry.
The Zariski topology of an
algebraic variety is the topology
whose closed sets are the algebraic subsets of the variety.
Over the field of complex numbers endowed with the usual metric
topology, all algebraic subsets are closed.
This goes to show that the Zarisky topology is coarser than the usual one.
The Zariski topology doesn't verify the Hausdorff
separation axiom (T2). However, the Fréchet axiom (T1) is verified: All finite sets are closed.