2019/03/20 by István Juhász, Juhász, István, Lajos Soukup +3
Mathematics · #54A25 #54A35 #54C10 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Mathematical Dynamics and Fractals #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1903.08532
openalex publication_date 2019/03/20 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We call a continuous map f : X \→ Y nowhere constant if it is not constant\non any non-empty open subset of its domain X. Clearly, this is equivalent\nwith the assumption that every fiber f-1(y) of f is nowhere dense in\nX. We call the continuous map f : X \→ Y pseudo-open if for each nowhere\ndense Z \⊂ Y its inverse image f-1(Z) is nowhere dense in X.\nClearly, if Y is crowded, i.e. has no isolated points, then f is nowhere\nconstant.\n The aim of this paper is to study the following, admittedly imprecise,\nquestion: How "small" nowhere constant, resp. pseudo-open continuous images can\n"large" spaces have? Our main results yield the following two precise answers\nto this question, explaining also our title. Both of them involve the cardinal\nfunction widehatc(X), the "hat version" of cellularity, which is defined\nas the smallest cardinal \κ such that there is no \κ-sized disjoint\nfamily of open sets in X. Thus, for instance, widehatc(X) = \ω1\nmeans that X is CCC.\n THEOREM A. Any crowded Tychonov space X has a crowded Tychonov nowhere\nconstant continuous image Y of weight w(Y) \≤ widehatc(X). Moreover, in\nthis statement \≤ may be replaced with < iff there are no\n widehatc(X)-Suslin lines (or trees).\n THEOREM B. Any crowded Tychonov space X has a crowded Tychonov pseudo-open\ncontinuous image Y of weight w(Y) \≤ 2^< widehatc(X). If Martin's\naxiom holds then there is a CCC crowded Tychonov space X such that for any\ncrowded Hausdorff pseudo-open continuous image Y of X we have w(Y) \≥\n mathfrakc ,( = 2< \ω1).\n