Geodesic
An observation on spaces with a zeroset diagonal
Mathematica Bohemica, Tome 145 (2020) no. 1, pp. 15-18
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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 $\mathfrak c$.
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 $\mathfrak c$.
DOI : 10.21136/MB.2018.0016-18
Classification : 54D20, 54E35
Keywords: first countable; discrete countable chain condition; zeroset diagonal; cardinal 
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Xuan, Wei-Feng. An observation on spaces with a zeroset diagonal. Mathematica Bohemica, Tome 145 (2020) no. 1, pp. 15-18. doi: 10.21136/MB.2018.0016-18

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