Topological spaces compact with respect to a set of filters

Abstract

If is a family of filters over some set I, a topological space X is sequencewise -compact if for every I-indexed sequence of elements of X there is such that the sequence has an F-limit point. Countable compactness, sequential compactness, initial κ-compactness, [λ; µ]-compactness, the Menger and Rothberger properties can all be expressed in terms of sequencewise -compactness for appropriate choices of . We show that sequencewise -compactness is preserved under taking products if and only if there is a filter such that sequencewise -compactness is equivalent to F-compactness. If this is the case, and there exists a sequencewise -compact T 1 topological space with more than one point, then F is necessarily an ultrafilter. The particular case of sequential compactness is analyzed in detail

Products of sequentially compact spaces and compactness with respect to a set of filters

“…On the other hand, a space is ω 1 -filtered if and only if in it every sequence converges. This is proved in Brandhorst [Br] or Brandhorst and Erné [BE], and can be also proved as in [L4,Lemma 4.1]. Since X is the product of a space in which every sequence converges and of a space satisfying the Rothberger property for countable covers, then Lemma 3.2 shows that X satisfies the Rothberger property for countable covers.…”

“…Then in [L3,Teorem 2.3] the Menger properties are explicitly described as sequencewise P-compactness, for some appropriate P consisting only of ultrafilters. That the Menger properties can be described as sequencewise P-compactness, for some P, follows directly from Theorem 2.7; the main point in [L3] is that the members of P can be chosen to be ultrafilters; this follows also abstractly from [L4,Corollary 5.3].…”

“…Notice that if in Corollary 6.4 we replace s by the rougher estimate c, then the corollary is a consequence of Theorem 3.1, since, by Remark 2.6, there is some P of cardinality c such that sequential compactness is equivalent to sequencewise P-compactness. As we mentioned in [L4,Problem 4.4], we do not know the value of the smallest cardinal ms such that sequential compactness is equivalent to sequencewise P-compactness, for some P with |P| = ms. Of course, if ms were equal to s, then Corollary 6.4 would be a direct consequence of Theorem 3.1. It follows from Remark 2.6 that ms ≤ c. Moreover, ms ≥ s, since otherwise, if ms < s, then by Theorem 3.1, we could prove Corollary 6.4 for the improved value ms in place of s, but we mentioned that s is the best possible value.…”

“…In [L4] we extended Kombarov notion to filters, and showed that this is a proper generalization, since, for example, sequential compactness can be characterized in terms of such a "local " filter convergence, but all the filters involved, in this case, are necessarily not maximal [L4,Section 5]. We called this general notion sequencewise P-compactness (here and in what follows P is always a family of filters over some set I) and in [L4] we mentioned that sequencewise P-compactness incorporates many compactness, covering and convergence properties, including sequential compactness, countable compactness, initial κ-compactness, [λ, µ]-compactness and the Menger and Rothberger properties.…”

Compactness of ω^λ for λ singular

“…So suppose that (3) fails as witnessed by some [λ, λ]-compact X = i∈I X i and an injective g : κ → I such that no X g(γ) is [µ γ , µ γ ]-compact. It is easy to see that if µ is regular then a space is not [µ, µ]-compact if and only if there is a continuous surjective function h : X → µ with the iit topology; see, e. g., Lipparini [13,Lemma 4]. Hence for each γ ∈ κ we have a continuous surjective function h γ : X γ → µ γ and, by naturality of products and since g is injective, a continuous surjective h : X → γ∈κ µ γ .…”