Zero-set ultrafilters and Gδ-closures in uniform spaces

ULTRAFILTERS AND COMPACTIFICATION OF UNIFORM SPACES í1)

In point set topology—or, more precisely, in the part of it once called "theory of abstract spaces"—there are two principal methods of investigation. The first one refers to the topological space alone(2) ; we shall call it the internal method : for example the separation axioms Po, Pi, T2 [A-H], the notions of regularity, normality, compactness, are expressed in terms of the topological space ...

Gδ-sets in topological spaces and games

Players One and Two play the following game: In the nth inning One chooses a set On from a prescribed family F of subsets of a space X; Two responds by choosing an open subset Tn of X. The players must obey the rule that On ⊆ On+1 ⊆ Tn+1 ⊆ Tn for each n. Two wins if the intersection of Two’s sets is equal to the union of One’s sets. If One has no winning strategy, then each element of F is a Gδ...

Limits in the Uniform Ultrafilters

Ultrafilters on G-spaces

For a discrete group G and a discrete G-space X, we identify the Stone-Čech compactifications βG and βX with the sets of all ultrafilters on G and X, and apply the natural action of βG on βX to characterize large, thick, thin, sparse and scattered subsets of X. We use G-invariant partitions and colorings to define G-selective and G-Ramsey ultrafilters on X. We show that, in contrast to the set-...

Gδ-separation axioms in ordered fuzzy topological spaces

The fuzzy concept has invaded all branches of Mathematics ever since the introduction of fuzzy set by Zadeh [10]. Fuzzy sets have applications in many fields such as information [5] and control [8]. The theory of fuzzy topological spaces was introduced and developed by Chang [3] and since then various notions in classical topology have been extended to fuzzy topological spaces. Sostak [6] intro...