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Keywords:
first countable; discrete countable chain condition; zeroset diagonal; cardinal
first countable; discrete countable chain condition; zeroset diagonal; cardinal
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 $\mathfrak c$.
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