A note on k-c-semistratifiable spaces and strong β -spaces

Li-Xia Wang; Liang-Xue Peng

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 3, page 287-299
  • ISSN: 0862-7959

Abstract

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Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are G δ -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T 2 -space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x X , { x } = { g ( x , n ) : n } and g ( x , n + 1 ) g ( x , n ) for each n . (2) If a sequence { x n } n of X converges to a point x X and y n g ( x n , n ) for each n , then for any convergent subsequence { y n k } k of { y n } n we have that { y n k } k converges to x . By the above characterization, we show that if X is a submesocompact locally k-c-semistratifiable space, then X is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. If X = { Int ( X n ) : n } and X n is a closed k-c-semistratifiable space for each n , then X is a k-c-semistratifiable space. In the last part of this note, we show that if X = { X n : n } and X n is a closed strong β -space for each n , then X is a strong β -space.

How to cite

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Wang, Li-Xia, and Peng, Liang-Xue. "A note on k-c-semistratifiable spaces and strong $\beta $-spaces." Mathematica Bohemica 136.3 (2011): 287-299. <http://eudml.org/doc/197240>.

@article{Wang2011,
abstract = {Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_\delta $-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\lbrace x\rbrace =\bigcap \lbrace g(x, n)\colon n\in \mathbb \{N\}\rbrace $ and $ g(x, n+1)\subseteq g(x, n)$ for each $n\in \mathbb \{N\}$. (2) If a sequence $\lbrace x_n\rbrace _\{n\in \mathbb \{N\}\}$ of $X$ converges to a point $x\in X$ and $y_n\in g(x_n, n)$ for each $n\in \mathbb \{N\}$, then for any convergent subsequence $\lbrace y_\{n_k\}\rbrace _\{k\in \mathbb \{N\}\}$ of $\lbrace y_n\rbrace _\{n\in \mathbb \{N\}\}$ we have that $\lbrace y_\{n_k\}\rbrace _\{k\in \mathbb \{N\}\}$ converges to $x$. By the above characterization, we show that if $X$ is a submesocompact locally k-c-semistratifiable space, then $X$ is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. If $X=\bigcup \lbrace \{\rm Int\}(X_n)\colon n\in \mathbb \{N\}\rbrace $ and $X_n$ is a closed k-c-semistratifiable space for each $n$, then $X$ is a k-c-semistratifiable space. In the last part of this note, we show that if $X=\bigcup \lbrace X_n\colon n\in \mathbb \{N\}\rbrace $ and $X_n$ is a closed strong $\beta $-space for each $n\in \mathbb \{N\}$, then $X$ is a strong $\beta $-space.},
author = {Wang, Li-Xia, Peng, Liang-Xue},
journal = {Mathematica Bohemica},
keywords = {c-semistratifiable space; k-c-semistratifiable space; submesocompact space; $g$ function; strong $\beta $-space; c-semistratifiable space; k-c-semistratifiable space; submesocompact space; g-function; strong -space},
language = {eng},
number = {3},
pages = {287-299},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on k-c-semistratifiable spaces and strong $\beta $-spaces},
url = {http://eudml.org/doc/197240},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Wang, Li-Xia
AU - Peng, Liang-Xue
TI - A note on k-c-semistratifiable spaces and strong $\beta $-spaces
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 3
SP - 287
EP - 299
AB - Recall that a space $X$ is a c-semistratifiable (CSS) space, if the compact sets of $X$ are $G_\delta $-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a $T_2$-space $X$ is a k-c-semistratifiable space if and only if $X$ has a $g$ function which satisfies the following conditions: (1) For each $x\in X$, $\lbrace x\rbrace =\bigcap \lbrace g(x, n)\colon n\in \mathbb {N}\rbrace $ and $ g(x, n+1)\subseteq g(x, n)$ for each $n\in \mathbb {N}$. (2) If a sequence $\lbrace x_n\rbrace _{n\in \mathbb {N}}$ of $X$ converges to a point $x\in X$ and $y_n\in g(x_n, n)$ for each $n\in \mathbb {N}$, then for any convergent subsequence $\lbrace y_{n_k}\rbrace _{k\in \mathbb {N}}$ of $\lbrace y_n\rbrace _{n\in \mathbb {N}}$ we have that $\lbrace y_{n_k}\rbrace _{k\in \mathbb {N}}$ converges to $x$. By the above characterization, we show that if $X$ is a submesocompact locally k-c-semistratifiable space, then $X$ is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. If $X=\bigcup \lbrace {\rm Int}(X_n)\colon n\in \mathbb {N}\rbrace $ and $X_n$ is a closed k-c-semistratifiable space for each $n$, then $X$ is a k-c-semistratifiable space. In the last part of this note, we show that if $X=\bigcup \lbrace X_n\colon n\in \mathbb {N}\rbrace $ and $X_n$ is a closed strong $\beta $-space for each $n\in \mathbb {N}$, then $X$ is a strong $\beta $-space.
LA - eng
KW - c-semistratifiable space; k-c-semistratifiable space; submesocompact space; $g$ function; strong $\beta $-space; c-semistratifiable space; k-c-semistratifiable space; submesocompact space; g-function; strong -space
UR - http://eudml.org/doc/197240
ER -

References

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  1. Bennett, H., Byerly, R., Lutzer, D., 10.1016/j.topol.2005.08.011, Topology Appl. 153 (2006), 2169-2181. (2006) Zbl1101.54034MR2239079DOI10.1016/j.topol.2005.08.011
  2. Borges, C. R., 10.2140/pjm.1966.17.1, Pacific J. Math. 17 (1966), 1-16. (1966) Zbl0175.19802MR0188982DOI10.2140/pjm.1966.17.1
  3. Creede, G. D., 10.2140/pjm.1970.32.47, Pacific J. Math. 32 (1970), 47-54. (1970) Zbl0189.23304MR0254799DOI10.2140/pjm.1970.32.47
  4. Engelking, R., General Topology, Sigma Series in Pure Mathematics 6, Heldermann, Berlin, revised ed. 1989. Zbl0684.54001MR1039321
  5. Gao, Z. M., On g -function separation, Questions Answers Gen. Topology 4 (1986), 47-57. (1986) Zbl0597.54027MR0852951
  6. Gao, Z. M., The closed images of metric spaces and Fréchet -spaces, Questions Answers Gen. Topology 5 (1987), 281-291. (1987) Zbl0643.54035MR0917886
  7. Good, C., Knight, R., Stares, I., 10.1016/S0166-8641(98)00128-X, Topology Appl. 101 (2000), 281-298. (2000) Zbl0938.54026MR1733809DOI10.1016/S0166-8641(98)00128-X
  8. Gruenhage, G., Generalized Metric Spaces. Handbook of Set-Theoretic Topology, North-Holland, Amsterdam (1984). (1984) MR0776629
  9. Hodel, R. E., 10.2140/pjm.1971.38.641, Pacific J. Math. 38 (1971), 641-652. (1971) MR0307169DOI10.2140/pjm.1971.38.641
  10. Kemoto, N., Yajima, Y., 10.1016/j.topol.2008.12.016, Topology Appl. 156 (2009), 1348-1354. (2009) Zbl1169.54003MR2502009DOI10.1016/j.topol.2008.12.016
  11. Kyung, B. L., 10.2140/pjm.1979.81.435, Pacific J. Math. 81 (1979), 435-446. (1979) MR0547610DOI10.2140/pjm.1979.81.435
  12. Lin, S., A note on k-semistratifiable spaces, J. Suzhou University (Natural Science) 4 (1988), 357-363. (1988) 
  13. Lin, S., Generalized Metric Spaces and Mappings, Chinese Science Publishers, Beijing (1995). (1995) MR1375020
  14. Lin, S., Mapping theorems on k-semistratifiable spaces, Tsukuba J. Math. 21 (1997), 809-815. (1997) Zbl1025.54501MR1603848
  15. Lutzer, D. J., 10.1016/0016-660X(71)90109-7, General Topology Appl. 1 (1971), 43-48. (1971) Zbl0211.25704MR0296893DOI10.1016/0016-660X(71)90109-7
  16. Martin, H. W., 10.4153/CJM-1973-086-0, Can. J. Math. 4 (1973), 840-841. (1973) Zbl0247.54031MR0328875DOI10.4153/CJM-1973-086-0
  17. Peng, L.-X., Wang, L. X., On C S S spaces and related conclusions, Chinese Acta Math. Sci. (Chin. Ser. A) 30 (2010), 358-363. (2010) Zbl1224.54065MR2664833
  18. Peng, L.-X., Lin, S., Monotone spaces and metrization theorems, Chinese Acta Math. Sinica (Chin. Ser.) 46 (2003), 1225-1232. (2003) Zbl1045.54010MR2035746
  19. Yajima, Y., Strong β -spaces and their countable products, Houston J. Math. 33 (2007), 531-540. (2007) Zbl1243.54046MR2308994

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