Reflections in small continuous images of ordered spaces

Abstract

We prove that countable i-weight reflects in continuous images of weight ≤ ω 1 for all Tychonoff spaces while separability reflects in continuous images of weight ≤ ω 1 for GO spaces. If X is a GO space and all continuous images of X of weight ≤ κ + have tightness at most κ, then t(X) ≤ κ. All continuous images of weight ≤ ω 1 of a GO space X have countable pseudocharacter if and only if X is hereditarily Lindelöf. Besides, all continuous images of weight ≤ ω 1 of a linearly ordered space X have G δ-diagonal if and only if X second countable.

On properties of compacta that do not reflect in small continuous images

“…We wish to thank Vladimir V. Tkachuk for sending us a preliminary version of [25] and for his suggestion to link up our result with some questions asked in [10]. We are also grateful to Marián Fabian, Witold Marciszewski and Roman Pol for several valuable comments.…”

“…The construction of the space is given in section 3 and uses the familiar idea of a ladder system associated to a given set S ⊆ ω 2 ; see, for instance, Pol [22] and Ciesielski and Pol [6] where a construction of this type was used to solve some problems on the structure of C(K) spaces. The existence of a stationary set in ω 2 with the above-mentioned properties is known to have an impact on other topological problems, see Fleissner [8], 3.10. In the framework of Banach space theory our result shows that it is relatively consistent that there exists a Banach space X of density ω 2 such that X is not weakly compactly generated while every subspace Y ⊆ X of density ≤ ω 1 is weakly compactly compactly generated, see section 4. In the final section of this note we give a partial negative answer to another problem from [25]. We show, assuming a weak version of Martin's axiom, that countable functional tightness does not reflect in small continuous images of compact spaces.…”

“…The following is an immediate consequence of a result due to Kamburelis [12], Lemma 3.1; see also [5], Theorem 4.4. We can now give a (partial) negative solution to Question 4.3 from [25].…”

“…Does a topological space X has a property (P) provided all its subspaces of small cardinality have property (P)?Tall [23] surveys results and problems of this type. Recently Tkachuk [24] and Tkachuk and Tkachenko [25] have investigated which topological properties reflect in small continuous images which, in particular, amounts to asking the following kind of questions.Problem 1.2. Does a topological space X has property (P) provided every continuous image of X of weight ≤ ω 1 has property (P)?Eberlein compacta and Corson compacta are two well-studied classes of compact spaces related to functional analysis, see the next section.…”

Reflecting properties in continuous images of small weight

“…Following the notation and terminology of [10,11,13], we say that a topological property P is reflectable in continuous images of weight κ in a class of topological spaces C if a space X ∈ C has the property P whenever every continuous image of X of weight at most κ has P. This paper is a sequel to [13] in which C is the class of generalized ordered (or GO-) spaces, that is, the class of all subspaces of ordered spaces. It was shown in [13] that in GO-spaces, unlike in the class of all Tychonoff spaces, density, network weight and tightness are all reflected in small continuous images. Among the properties considered here are that of being κ-monolithic, having countable extent, pseudocompactness and countable compactness.…”