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.
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.
We show that a regular totally ω-narrow paratopological group G has countable index of regularity, i.e., for every neighborhood U of the identity e of G, we can find a neighborhood V of e and a countable family of neighborhoods of e in G such that ∩W∈γ VW−1⊆ U. We prove that every regular (Hausdorff) totally !-narrow paratopological group is completely regular (functionally Hausdorff). We show that the index of regularity of a regular paratopological group is less than or equal to the weak Lindelöf...
We examine when a space has a zero set universal parametrised by a metrisable space of minimal weight and show that this depends on the -weight of when is perfectly normal. We also show that if parametrises a zero set universal for then for all . We construct zero set universals that have nice properties (such as separability or ccc) in the case where the space has a -coarser topology. Examples are given including an space with zero set universal parametrised by an space (and...
Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put . We show that f ∈ (α) iff for some natural number n there are infinite cardinals and ordinals such that and where each . Under GCH we prove that if α < ω₂ then (i) ; (ii) if λ > cf(λ) = ω, ; (iii) if cf(λ) = ω₁, ; (iv) if cf(λ) > ω₁, . This yields a complete characterization of the classes (α) for all α < ω₂,...
A topological space has a rank 2-diagonal if there exists a diagonal sequence on of rank , that is, there is a countable family of open covers of such that for each , . We say that a space satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of is countable. We mainly prove that if is a DCCC normal space with a rank 2-diagonal, then the cardinality of is at most . Moreover, we prove that if is a first countable...
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...
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].
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.
We discuss various generalizations of the class of Lindelöf spaces and study the difference between two of these generalizations, the classes of star-Lindelöf and centered-Lindelöf spaces.