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Reflecting properties in continuous images of small weight. (English) Zbl 1362.54012

Let \(\kappa\) be an infinite cardinal and let \(\mathcal{P}\) be a topological property of spaces from a class \(\mathcal{C}\) of Tychonoff spaces. The property \(\mathcal{P}\) is called reflectable in continuous images of weight \(\kappa\) in the class \(\mathcal{C}\) if a space \(X \in \mathcal{C}\) has property \(\mathcal{P}\) whenever for every continuous surjection \(f : X\to Y\) with \(w(Y) \leq \kappa\), the space \(Y\) has the property \(\mathcal{P}\).
The paper under review is a sequel of the paper “Reflections in small continuous images of ordered spaces” published in the same volume by V. V. Tkachuk and R. G. Wilson [Eur. J. Math. 2, No. 2, 508–517 (2016; Zbl 1344.54001)]. In this paper reflectability was considered in the class of generalized ordered spaces. Among properties investigated in the present paper there are density, network weight, tightness, monolithicity and extent in the class of GO-spaces, pseudocompact spaces and perfectly normal spaces.
The authors obtain several interesting results and answer a number of questions posed in the previous paper. In particular it is shown that the property of being feebly \(\kappa\)-Lindelöf is reflected in continuous images of weight \(\kappa^+\). Investigations of GO-spaces from the point of view of reflecting properties lead to a theorem which is interesting in its own right. Namely, in the paper it is proved that if a GO-space has extent not less than \(\kappa^+\) then it can be mapped by a continuous mapping onto the discrete space of cardinality \(\kappa^+\). It is also proved that a Tychonoff space \(X\) has the property that every continuous image of \(X\) of weight not exceeding \(\omega_1\) has countable pseudocharacter iff \(X\) is compact and every closed subset of \(X\) is a \(G_\delta\)-set. The authors raise the question of whether it is true in ZFC that a countably compact space of countable projective tightness has countable tightness.

MSC:

54C05 Continuous maps
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)

Citations:

Zbl 1344.54001
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References:

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