-compactness modulo an ideal.
Cardinal invariants of -topologies.
Cardinalities of DCCC normal spaces with a rank 2-diagonal
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...
Čech cohomology and covering dimension for topological spaces
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...
Centered-Lindelöfness versus star-Lindelöfness
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.
Characterizations of generalized paracompact spaces
C(K) spaces which cannot be uniformly embedded into c₀(Γ)
We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c₀(Γ) for any set Γ. The first one is [0,ω₁] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω₀ + 1.
Cleavability and divisibility over developable spaces
Some results on cleavability theory are presented. We also show some new [16]'s results.
Closed discrete subsets of separable spaces and relative versions of normality, countable paracompactness and property
In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property . It follows that it is consistent that closed discrete subsets of a separable space which are also relatively normal (relatively countably paracompact, relatively ) in are necessarily countable. There are, however, consistent examples of...
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.
Closed subsets of star -compact spaces
Closure-preserving covers
Closure-preserving families of finite sets
Collectionwise Hausdorff property in product spaces
Combinatorics of open covers (III): games, Cp (X)
Some of the covering properties of spaces as defined in Parts I and II are here characterized by games. These results, applied to function spaces of countable tightness, give new characterizations of countable fan tightness and countable strong fan tightness. In particular, each of these properties is characterized by a Ramseyan theorem.
Combinatorics of open covers (VII): Groupability
We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a -space. In [9] we showed that has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. has countable fan tightness and the Reznichenko property. 2....
Compactification-like extensions [Book]
Compact-like properties in hyperspaces
Compactness of Powers of ω
We characterize exactly the compactness properties of the product of κ copies of the space ω with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard elements in elementary extensions. We also have results involving products of possibly uncountable regular cardinals.