Article
Full entry |
PDF
(0.2 MB)
Feedback
Keywords:
non-normality point; butterfly-point; nice family; nice space; metrizable crowded space; Sorgenfrey line
non-normality point; butterfly-point; nice family; nice space; metrizable crowded space; Sorgenfrey line
Summary:
J. Terasawa in "$\beta X-\{p\}$ are non-normal for non-discrete spaces $X$" (2007) and the author in "On non-normality points and metrizable crowded spaces" (2007), independently showed for any metrizable crowded space $X$ that each point $p$ of its Čech--Stone remainder $X^{*}$ is a non-normality point of $\beta X$. We introduce a new class of spaces, named nice spaces, which contains both of Sorgenfrey line and every metrizable crowded space. We obtain the result above for every nice space.
References:
[1] Błaszczyk A., Szymański A.: Some non-normal subspaces of the Čech–Stone compactification of a discrete space. in Abstracta, 8th Winter School on Abstract Analysis, Czechoslovak Academy of Sciences, Praha, 1980, pages 35–38.
[2] Logunov S.: On hereditary normality of compactifications. Topology Appl. 73 (1996), no. 3, 213–216. DOI 10.1016/S0166-8641(96)00063-6
[3] Logunov S.: On hereditary normality of zero-dimensional spaces. Topology Appl. 102 (2000), no. 1, 53–58. DOI 10.1016/S0166-8641(98)00137-0
[4] Logunov S.: On remote points, non-normality and $\pi $-weight $\omega _{1}$. Comment. Math. Univ. Carolin. 42 (2001), no. 2, 379–384.
[5] Logunov S.: On non-normality points and metrizable crowded spaces. Comment. Math. Univ. Carolin. 48 (2007), no. 3, 523–527.
[6] Logunov S.: Non-normality points and big products of metrizable spaces. Topology Proc. 46 (2015), 73–85.
[7] Šapirovskiĭ B. È.: The embedding of extremely disconnected spaces in bicompacta. b-points and weight of point-wise normal spaces. Dokl. Akad. Nauk SSSR 223 (1975), no. 5, 1083–1086 (Russian).
[8] Terasawa J.: $\beta X-\{p\}$ are non-normal for non-discrete spaces $X$. Topology Proc. 31 (2007), no. 1, 309–317.
[9] Warren N. M.: Properties of Stone–Čech compactifications of discrete spaces. Proc. Amer. Math. Soc. 33 (1972), 599–606.
Search
Partner of