Amer Bešlagić

Normality in products

Abstract If P is a paracompact p -space, P × X is collectionwise normal, and Y is a closed image of X , then P × Y is collectionwise normal. If M is metric, X shrinking, and M × X is normal, then M × X is shrinking.

Set-theoretic constructions of nonshrinking open covers

Abstract A family { M α | αϵA } is a shrinking of a cover { O α | αϵA } of a topological space if { M α | αϵA } also covers and M α ⊂ O α for all αϵA . ◊ ++ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking. ◊ implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.

Yet another Dowker product

Abstract ♦ implies that there is a perfectly normal X such that X × X is normal but not countably paracompact.

Spaces of nonuniform ultrafilters in spaces of uniform ultrafilters

Abstract For every cardinal κ ⩾ ω , the following is consistent: each x in U ( κ ) (the space of uniform ultrafilters on κ) is a nonnormality point of U ( κ ), i.e., U ( κ )⧹{ x } is not normal, because the space of nonuniform ultrafilters on cf(2 κ ) embeds as a closed subspace in U ( κ )⧹{ x }.

A hereditarily normal square

Abstract ♦ implies that there is a countably compact noncompact space X so that X × X is hereditarily normal. This shows that the following statement is independent from ZFC: Every countably compact X with X × X hereditarily normal is compact.

The normality of products with one compact factor, revisited

Abstract A simpler proof of the following theorem by Rudin is given: If C is compact, X × C normal, and Y a closed image of X , then Y × C is normal.

Partitions of vector spaces

The following question is answered: if the real line is partitioned into countable sets, is there a Hamel basis that pick at most one element from each member of the partition?