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𝒫 -approximable compact spaces

Mihail G. Tkachenko (1991)

Commentationes Mathematicae Universitatis Carolinae

For every topological property 𝒫 , we define the class of 𝒫 -approximable spaces which consists of spaces X having a countable closed cover γ such that the “section” X ( x , γ ) = ⋂ { F ∈ γ : x ∈ F } has the property 𝒫 for each x ∈ X . It is shown that every 𝒫 -approximable compact space has 𝒫 , if 𝒫 is one of the following properties: countable tightness, ℵ 0 -scatteredness with respect to character, C -closedness, sequentiality (the last holds under MA or 2 ℵ 0 < 2 ℵ 1 ). Metrizable-approximable spaces are studied: every compact space in this class has...

A compact Hausdorff topology that is a T₁-complement of itself

Dmitri Shakhmatov, Michael Tkachenko (2002)

Fundamenta Mathematicae

Topologies τ₁ and τ₂ on a set X are called T₁-complementary if τ₁ ∩ τ₂ = X∖F: F ⊆ X is finite ∪ ∅ and τ₁∪τ₂ is a subbase for the discrete topology on X. Topological spaces ( X , τ X ) and ( Y , τ Y ) are called T₁-complementary provided that there exists a bijection f: X → Y such that τ X and f - 1 ( U ) : U ∈ τ Y are T₁-complementary topologies on X. We provide an example of a compact Hausdorff space of size 2 which is T₁-complementary to itself ( denotes the cardinality of the continuum). We prove that the existence of a compact Hausdorff...

A construction of a Fréchet-Urysohn space, and some convergence concepts

Aleksander V. Arhangel'skii (2010)

Commentationes Mathematicae Universitatis Carolinae

Some strong versions of the Fréchet-Urysohn property are introduced and studied. We also strengthen the concept of countable tightness and generalize the notions of first-countability and countable base. A construction of a topological space is described which results, in particular, in a Tychonoff countable Fréchet-Urysohn space which is not first-countable at any point. It is shown that this space can be represented as the image of a countable metrizable space under a continuous pseudoopen mapping....

A Corson compact L-space from a Suslin tree

Peter Nyikos (2015)

Colloquium Mathematicae

The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.

A Dowker group

Klaas Pieter Hart, Heikki J. K. Junnila, Jan van Mill (1985)

Commentationes Mathematicae Universitatis Carolinae

A game and its relation to netweight and D-spaces

Gary Gruenhage, Paul Szeptycki (2011)

Commentationes Mathematicae Universitatis Carolinae

We introduce a two player topological game and study the relationship of the existence of winning strategies to base properties and covering properties of the underlying space. The existence of a winning strategy for one of the players is conjectured to be equivalent to the space have countable network weight. In addition, connections to the class of D-spaces and the class of hereditarily Lindelöf spaces are shown.

A generic theorem in the theory of cardinal invariants of topological spaces

Aleksander V. Arhangel'skii (1995)

Commentationes Mathematicae Universitatis Carolinae

Relative versions of many important theorems on cardinal invariants of topological spaces are formulated and proved on the basis of a general technical result, which provides an algorithm for such proofs. New relative cardinal invariants are defined, and open problems are discussed.

A new Lindelöf space with points G δ

Alan S. Dow (2015)

Commentationes Mathematicae Universitatis Carolinae

We prove that ♦ * implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality 2 ℵ 1 which has points G δ . In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

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