Classifying stationary sets: a survey
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...
Closure, interior, and union in finite topological spaces
Cofinal families in certain function spaces
Collectionwise Hausdorffness at limit cardinals
Coloring Cantor sets and resolvability of pseudocompact spaces
Let us denote by the statement that , i.e. the Baire space of weight , has a coloring with colors such that every homeomorphic copy of the Cantor set in picks up all the colors. We call a space -regular if it is Hausdorff and for every nonempty open set in there is a nonempty open set such that . We recall that a space is called feebly compact if...
Combinatorial aspects of measure and category
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....
Compact sets without converging sequences in the random real model.
Compacta are maximally -resolvable
It is well-known that compacta (i.e. compact Hausdorff spaces) are maximally resolvable, that is every compactum contains many pairwise disjoint dense subsets, where denotes the minimum size of a non-empty open set in . The aim of this note is to prove the following analogous result: Every compactum contains many pairwise disjoint -dense subsets, where denotes the minimum size of a non-empty set in .
Compact-calibres of regular and monotonically normal spaces.
Compact-covering numbers
Compactification and the continuum hypothesis
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.
Concerning certain minimal cover refineable spaces
Connected LCA groups are sequentially connected
We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if is a connected locally compact Abelian subgroup of a Hausdorff topological group and the quotient space is sequentially connected, then so is .
Constructions of thin-tall Boolean spaces.
This is an expository paper about constructions of locally compact, Hausdorff, scattered spaces whose Cantor-Bendixson height has cardinality greater than their Cantor-Bendixson width.
Continuous functions on products of topological spaces
Continuous images of Lindelöf -groups, -compact groups, and related results
It is shown that there exists a -compact topological group which cannot be represented as a continuous image of a Lindelöf -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space is a continuous image of a Lindelöf -group, then there exists a covering of by dyadic compacta such that . We also show that if a homogeneous compact space is a continuous...
Convergent sequences in