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Reflecting topological properties in continuous images. (English) Zbl 1257.54022

Let \(\kappa\) be an infinite cardinal. A cardinal invariant \(\eta\) is said to reflect in continuous images of weight \(\leq \kappa^+\) if for any space \(X\) to satisfy \(\eta(X)\leq\kappa\) it is enough that \(\eta(Y)\leq\kappa\) for every continuous image \(Y\) of \(X\) with weight \(\leq \kappa^+\). Analogously, a topological property \(\mathcal P\) is said to reflect in continuous images of weight \(\leq \kappa^+\) if the following are equivalent for any space \(X\): (i) \(X\) has \(\mathcal P\). (ii) Every continuous image of \(X\) with weight less than or equal to \(\kappa^+\) has \(\mathcal P\).
From the Introduction: “We establish that, for any infinite cardinal \(\kappa\), the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight \(\kappa^+\) for arbitrary Tychonoff spaces; the same is true for network weight under the hypothesis \(2^{\kappa}=\kappa^+\). We also show that tightness reflects in continuous images of weight \(\kappa^+\) for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight \(\omega_1.\)”
Further results are proved under additional axioms of set theory and a number of examples and open questions are provided.

MSC:

54C05 Continuous maps
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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