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Some examples related to colorings. (English) Zbl 1050.54023
Summary: We complement the literature by proving that for a fixed-point free map \(f: X \to X\) the statements (1) \(f\) admits a finite functionally closed cover \(\mathcal A\) with \(f[A] \cap A =\emptyset \) for all \(A \in \mathcal A\) (i.e., a coloring) and (2) \(\beta f\) is fixed-point free are equivalent.
When functionally closed is weakened to closed, we show that normality is sufficient to prove equivalence, and give an example to show it cannot be omitted.
We also show that a theorem due to Jan van Mill is sharp: for every \(n \geq 2\) we construct a strongly zero-dimensional Tikhonov space \(X\) and a fixed-point free map \(f: X \to X\) such that \(f\) admits a closed coloring, but no coloring has cardinality less than \(n\).
MSC:
54G20 Counterexamples in general topology
54C20 Extension of maps
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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