An observation on spaces with a zeroset diagonal
Mathematica Bohemica (2020)
- Volume: 145, Issue: 1, page 15-18
- ISSN: 0862-7959
Access Full Article
top
Abstract
topHow to cite
topXuan, Wei-Feng. "An observation on spaces with a zeroset diagonal." Mathematica Bohemica 145.1 (2020): 15-18. <http://eudml.org/doc/297119>.
@article{Xuan2020,
abstract = {We say that a space $X$ has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. A space $X$ has a zeroset diagonal if there is a continuous mapping $f\colon X^2 \rightarrow [0,1]$ with $\Delta _X=f^\{-1\}(0)$, where $\Delta _X=\lbrace (x,x)\colon x\in X\rbrace $. In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most $\mathfrak \{c\}$.},
author = {Xuan, Wei-Feng},
journal = {Mathematica Bohemica},
keywords = {first countable; discrete countable chain condition; zeroset diagonal; cardinal},
language = {eng},
number = {1},
pages = {15-18},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An observation on spaces with a zeroset diagonal},
url = {http://eudml.org/doc/297119},
volume = {145},
year = {2020},
}
TY - JOUR
AU - Xuan, Wei-Feng
TI - An observation on spaces with a zeroset diagonal
JO - Mathematica Bohemica
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 145
IS - 1
SP - 15
EP - 18
AB - We say that a space $X$ has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. A space $X$ has a zeroset diagonal if there is a continuous mapping $f\colon X^2 \rightarrow [0,1]$ with $\Delta _X=f^{-1}(0)$, where $\Delta _X=\lbrace (x,x)\colon x\in X\rbrace $. In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most $\mathfrak {c}$.
LA - eng
KW - first countable; discrete countable chain condition; zeroset diagonal; cardinal
UR - http://eudml.org/doc/297119
ER -
References
top- Arhangel'skii, A. V., Buzyakova, R. Z., The rank of the diagonal and submetrizability, Commentat. Math. Univ. Carol. 47 (2006), 585-597. (2006) Zbl1150.54335MR2337413
- Buzyakova, R. Z., Observations on spaces with zeroset or regular -diagonals, Commentat. Math. Univ. Carol. 46 (2005), 469-473. (2005) Zbl1121.54051MR2174525
- Buzyakova, R. Z., 10.1016/j.topol.2005.06.004, Topology Appl. 153 (2006), 1696-1698. (2006) Zbl1094.54001MR2227022DOI10.1016/j.topol.2005.06.004
- Engelking, R., General Topology, Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989). (1989) Zbl0684.54001MR1039321
- Ginsburg, J., Woods, R. G., 10.2307/2041457, Proc. Am. Math. Soc. 64 (1977), 357-360. (1977) Zbl0398.54002MR0461407DOI10.2307/2041457
- Gotchev, I. S., Cardinalities of weakly Lindelöf spaces with regular -diagonals, Available at https://arxiv.org/abs/1504.01785 (2015). (2015) MR3958260
- Hodel, R. E., 10.1016/c2009-0-12309-7, Handbook of Set-Theoretic Topology North-Holland, Amsterdam (1984), 1-61 K. Kunen et al. (1984) Zbl0559.54003MR0776620DOI10.1016/c2009-0-12309-7
- Shakhmatov, D., No upper bound for cardinalities of Tychonoff c.c.c. spaces with a -diagonal exists. An answer to J. Ginsburg and R. G. Woods’ question, Commentat. Math. Univ. Carol. 25 (1984), 731-746. (1984) Zbl0572.54003MR0782022
- Uspenskij, V. V., A large -discrete Fréchet space having the Souslin property, Commentat. Math. Univ. Carol. 25 (1984), 257-260. (1984) Zbl0553.54001MR0768812
- Wage, M. L., Fleissner, W. G., Reed, G. M., 10.1090/S0002-9904-1976-14150-X, Bull. Am. Math. Soc. 82 (1976), 635-639. (1976) Zbl0332.54018MR0410665DOI10.1090/S0002-9904-1976-14150-X
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.