Nearly Countable Dense Homogeneous Spaces

AbstractWe study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X ∖ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or c many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.

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Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

Journal of Mathematical Logic ◽

10.1142/s0219061320500154 ◽

2020 ◽

Vol 20 (03) ◽

pp. 2050015

Author(s):

Raphaël Carroy ◽

Andrea Medini ◽

Sandra Müller

Keyword(s):

Homogeneous Space ◽

General Result ◽

Homogeneous Spaces ◽

Locally Compact ◽

Compact Spaces ◽

Axiom Of Determinacy ◽

Locally Compact Spaces

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

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General function spaces, products and continuous lattices

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066020 ◽

1986 ◽

Vol 100 (2) ◽

pp. 193-205 ◽

Cited By ~ 11

Author(s):

John Isbell

Keyword(s):

General Result ◽

Metric Spaces ◽

Function Spaces ◽

Partial Result ◽

General Function ◽

Compact Open Topology ◽

Locally Compact ◽

Compact Spaces ◽

Continuous Lattices ◽

Locally Compact Spaces

The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

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On 𝒞ℛ-Epic Embeddings and Absolute 𝒞ℛ-Epic Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2005-043-2 ◽

2005 ◽

Vol 57 (6) ◽

pp. 1121-1138 ◽

Cited By ~ 1

Author(s):

Michael Barr ◽

R. Raphael ◽

R. G. Woods

Keyword(s):

Locally Compact ◽

Compact Spaces ◽

Open Questions ◽

Tychonoff Spaces ◽

Locally Compact Spaces

AbstractWe study Tychonoff spaces X with the property that, for all topological embeddings X → Y, the induced map C(Y ) → C(X) is an epimorphism of rings. Such spaces are called absolute 𝒞ℛ-epic. The simplest examples of absolute 𝒞ℛ-epic spaces are σ-compact locally compact spaces and Lindelöf P-spaces. We show that absolute CR-epic first countable spaces must be locally compact.However, a “bad” class of absolute CR-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute CR-epic abound, and some are presented.

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