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

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 \(\mathfrak{c}\).


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