Mathematica Bohemica, Vol. 145, No. 1, pp. 15-18, 2020
An observation on spaces with a zeroset diagonal
Wei-Feng Xuan
Received February 6, 2018. Published online November 26, 2018.
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 $f X^2 \rightarrow[0,1]$ 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 $\mathfrak c$.
Keywords: first countable; discrete countable chain condition; zeroset diagonal; cardinal
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Affiliations: Wei-Feng Xuan, School of Statistics and Mathematics, Nanjing Audit University, 86 West Yushan Road, Pukou District, Nanjing, 211815, China, e-mail: wfxuan@nau.edu.cn
