An observation on spaces with a zeroset diagonal. (English) Zbl 1474.54075

Xuan, Wei-Feng

Math. Bohem. 145, No. 1, 15-18 (2020).

Summary: 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 \(\Delta_X=\{(x,x)\colon x\in X\}\). In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most \(\mathfrak{c}\).

MSC:

54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)
54E35	Metric spaces, metrizability
54A25	Cardinality properties (cardinal functions and inequalities, discrete subsets)

Keywords:

first countable; discrete countable chain condition; zeroset diagonal; cardinal

References:

[1]	Arhangel’skii, A. V.; Buzyakova, R. Z., The rank of the diagonal and submetrizability, Commentat. Math. Univ. Carol. 47 (2006), 585-597 · Zbl 1150.54335
[2]	Buzyakova, R. Z., Observations on spaces with zeroset or regular \(G_\delta \)-diagonals, Commentat. Math. Univ. Carol. 46 (2005), 469-473 · Zbl 1121.54051
[3]	Buzyakova, R. Z., Cardinalities of ccc-spaces with regular \(G_\delta \)-diagonals, Topology Appl. 153 (2006), 1696-1698 · Zbl 1094.54001
[4]	Engelking, R., General Topology, Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989) · Zbl 0684.54001
[5]	Ginsburg, J.; Woods, R. G., A cardinal inequality for topological spaces involving closed discrete sets, Proc. Am. Math. Soc. 64 (1977), 357-360 · Zbl 0398.54002
[6]	Gotchev, I. S., Cardinalities of weakly Lindelöf spaces with regular \(G_\kappa \)-diagonals, Available at https://arxiv.org/abs/1504.01785 (2015) · Zbl 1426.54009
[7]	Hodel, R. E., Cardinal function. I, Handbook of Set-Theoretic Topology North-Holland, Amsterdam (1984), 1-61 K. Kunen et al · Zbl 0559.54003
[8]	Shakhmatov, D., No upper bound for cardinalities of Tychonoff c.c.c. spaces with a \(G_\delta \)-diagonal exists. An answer to J. Ginsburg and R. G. Woods’ question, Commentat. Math. Univ. Carol. 25 (1984), 731-746 · Zbl 0572.54003
[9]	Uspenskij, V. V., A large \(F_{\sigma}\)-discrete Fréchet space having the Souslin property, Commentat. Math. Univ. Carol. 25 (1984), 257-260 · Zbl 0553.54001
[10]	Wage, M. L.; Fleissner, W. G.; Reed, G. M., Normality versus countable paracompactness in perfect spaces, Bull. Am. Math. Soc. 82 (1976), 635-639 · Zbl 0332.54018