Ultrafilter invariants in topological spaces

Abstract

Abstract. Let m > n0 and X = ]l¡elX¡. Then X is [n0, m]-compact if and only if ¡¡¡eJX¡ is [K0, m]-compact for all J c / with |y| < 22™. Let m > Kq, 0{: £ < m) a net in X,p e X, and <$ e /3(m). Then/? = «¡) -lim£<m x£ if {£ < m: x{ 6 £/} e ¿D for every neighborhood U of p. Every topological space is characterized by its ^D-limits. Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then A-is a continuous perfect image of a subspace of ßD which contains D if and only if A" is regular.

Compactness of Powers of ω

“…The following observation by Saks [Sa,, building also on ideas of Bernstein and Ginsburg, will play a fundamental role in the present note. We shall assume that λ is regular, so that we do not need the assumption that sequences are faithfully indexed and, moreover, as wellknown, in this case, rλ, λs-compactness is equivalent to the statement that every subset of cardinality λ has a complete accumulation point (Crλ, λs in Saks' notation).…”

The 𝛼-boundification of 𝛼

“…For other results on spaces required to be p-compact simultaneously for various p, see Woods [18] and Saks [17].…”

The non-equivalence of τ-ultracompactness and τ-boundedness

“…V. Saks [2] generalizes the notion of a p-limit point to transfinite sequences in the following way: let τ be an infinite cardinal; if p ∈ βτ \ τ is a free ultrafilter on τ (with the discrete topology) and (x α : α ∈ τ) (for short (x α )) is a τ-sequence in a space X, then x ∈ X is a p-limit point of (x α ), denoted by x = p − lim x α , if for each neighborhood U of x, {α : x α ∈ U} ∈ p and we can say, in this case, that (x α ) p-converges to x. Saks further extends p-compactness for any ultrafilter p ∈ βτ \ τ where a space X is p-compact if any τ-sequence in X has a p-limit point. He proves there that in the class of regular spaces the notions of τ-boundedness and τ-ultracompactness are equivalent for any infinite cardinal τ, where τ-boundedness means that the closure of any subset of cardinality not exceeding τ is compact and τ-ultracompactness means that X is p-compact for any p ∈ βτ \ τ.…”