Cardinal invariants and compactifications
We prove that every compact space is a Čech-Stone compactification of a normal subspace of cardinality at most , and some facts about cardinal invariants of compact spaces.
Categories of topological spaces with sufficiently many sequentially closed spaces
Categories of Wallman extendible functions
Cauchy Spaces. II. Regular Completions and Compactifications.
Čech complete nearness spaces
We study Čech complete and strongly Čech complete topological spaces, as well as extensions of topological spaces having these properties. Since these two types of completeness are defined by means of covering properties, it is quite natural that they should have a convenient formulation in the setting of nearness spaces and that in that setting these formulations should lead to new insights and results. Our objective here is to give an internal characterization of (and to study) those nearness...
Čech-completeness and ultracompleteness in “nice spaces”
We prove that if is a union of subspaces of pointwise countable type then the space is of pointwise countable type. If is a countable union of ultracomplete spaces, the space is ultracomplete. We give, under CH, an example of a Čech-complete, countably compact and non-ultracomplete space, giving thus a partial answer to a question asked in [BY2].
Čech-Stone-like compactifications for general topological spaces
The problem whether every topological space has a compactification such that every continuous mapping from into a compact space has a continuous extension from into is answered in the negative. For some spaces such compactifications exist.
Čechův-Stoneův -obal
Circle spaces
Closed embeddings into complements of -products
In some sense, a dual property to that of Valdivia compact is considered, namely the property to be embedded as a closed subspace into a complement of a -subproduct of a Tikhonov cube. All locally compact spaces are co-Valdivia spaces (and only those among metrizable spaces or spaces having countable type). There are paracompact non-locally compact co-Valdivia spaces. A possibly new type of ultrafilters lying in between P-ultrafilters and weak P-ultrafilters is introduced. Under Martin axiom and...
Closed mappings and the Freudenthal compactification
Closed subsets of absolutely star-Lindelöf spaces II
In this paper, we prove the following two statements: (1) There exists a discretely absolutely star-Lindelöf Tychonoff space having a regular-closed subspace which is not CCC-Lindelöf. (2) Every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented in a Hausdorff (regular, Tychonoff) absolutely star-Lindelöf space as a closed subspace.
Coarse dimensions and partitions of unity.
Gromov and Dranishnikov introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov and Dranishnikov-Keesling-Uspienskij.
Coarse structures and group actions
The main results of the paper are: Proposition 0.1. A group G acting coarsely on a coarse space (X,𝓒) induces a coarse equivalence g ↦ g·x₀ from G to X for any x₀ ∈ X. Theorem 0.2. Two coarse structures 𝓒₁ and 𝓒₂ on the same set X are equivalent if the following conditions are satisfied: (1) Bounded sets in 𝓒₁ are identical with bounded sets in 𝓒₂. (2) There is a coarse action ϕ₁ of a group G₁ on (X,𝓒₁) and a coarse action ϕ₂ of a...
Compactification and compactoidification
Compactification and the continuum hypothesis
Compactification of pointed 1-movable spaces
Compactification of Products
Compactification of the domains of certain rings of functions