Compactness of ω^λ for λ singular

Abstract

We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.2010 Mathematics Subject Classification 54B10, 54D20, 03C75 (primary); 03C20, 03E05, 54A20 (secondary)

Topological spaces compact with respect to a set of filters

“…Moreover, all the above arguments apply to [µ, λ]-compactness, too; see the next section. Other results about preservation of [µ, λ]-compactness under products are given in [17], where we establish a connection with strongly compact cardinals.…”

A note on λ-compact spaces

“…We should remind the reader that our notion of λ-compactness coincides with the concept of Lindelöf + cardinal of a topological space, see [10]. In fact when a space X is λ-compact, then λ is the least cardinal such that X is a finally λ-compact space, see [9], [10].…”

“…In fact when a space X is λ-compact, then λ is the least cardinal such that X is a finally λ-compact space, see [9], [10]. The reader is referred to [10], where it is shown that the notion of ordinal compactness is much more applicable than cardinal compactness.…”

“…In fact when a space X is λ-compact, then λ is the least cardinal such that X is a finally λ-compact space, see [9], [10]. The reader is referred to [10], where it is shown that the notion of ordinal compactness is much more applicable than cardinal compactness. Moreover, in [10] it is also rightly noted that the concept of λ-compactness is more convenient, since it distinguishes between compactness and Lindelöfness.…”

“…The reader is referred to [10], where it is shown that the notion of ordinal compactness is much more applicable than cardinal compactness. Moreover, in [10] it is also rightly noted that the concept of λ-compactness is more convenient, since it distinguishes between compactness and Lindelöfness. We note that compact spaces (respectively, Lindelöf noncompact spaces) are ℵ • -compact (respectively, ℵ 1 -compact spaces) and in general every topological space X is λ-compact for some infinite cardinal number λ, see the concept of Lindelöf number in [2: p. 193], see also [6: Definition 1.8].…”