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Abstrakty
Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T₅ spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theorem (Theorem 1) that applies to all such ω₁-compact spaces and another (Theorem 4) to all such spaces of Lindelöf number ≤ ℵ₁. It also introduces an axiom (Axiom F) on crowding of functions, with consequences (Theorem 3) for the crowding of countably compact subspaces in certain continuous preimages of ω₁. It also exposes (Theorem 2) the fine structure of perfect preimages of ω₁ which are T₅ and hereditarily collectionwise Hausdorff. In these theorems, "T₅ and hereditarily collectionwise Hausdorff" is weakened to "hereditarily strongly collectionwise Hausdorff." Corollaries include the consistency, modulo large cardinals, of every hereditarily strongly collectionwise Hausdorff manifold of dimension > 1 being metrizable. The concept of an alignment plays an important role in formulating several of the structure theorems.
Słowa kluczowe
Kategorie tematyczne
- 03E50: Continuum hypothesis and Martin's axiom
- 03E55: Large cardinals
- 54D30: Compactness
- 54A35: Consistency and independence results
- 54F35: Higher-dimensional local connectedness
- 54D05: Connected and locally connected spaces (general aspects)
- 03E35: Consistency and independence results
- 54D45: Local compactness, σ -compactness
- 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
Czasopismo
Rocznik
Tom
Numer
Strony
25-45
Opis fizyczny
Daty
wydano
2003
Twórcy
autor
- Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.
Bibliografia
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-1-3