The equivalence of two definitions of sequential pseudocompactness

Abstract

We show that two possible definitions of sequential pseudocompactness are equivalent, and point out some consequences.2010 MSC: Primary 54D20; Secondary 54B10.

On old and new classes of feebly compact spaces

“…2 One of the authors introduced this notion a few years ago as a natural property intermediate between feeble and sequential compactness, which may be useful in some applications in topological algebra. Indeed, for instance, Proposition 1.10. by Artico et al [4] combined with Theorem 1.1 by Lipparini [17] states that that each T 0 feebly compact topological group is sequentially feebly compact. But later we found that it is a known property, even with the same name.…”

“…In general topology we often investigate different classes of compact-like spaces and relations between them, see, for instance, basic [11,Chap. 3] and general works [9], [19], [23], [22], [17]. We consider the present paper as a next small step in this quest.…”

“…1.8], where are used pairwise disjoint open sets instead. Lipparini proved in[17] that these notions are equivalent.…”

Maximalidad de propiedades que generalizan la compacidad

“…También, S. García-Ferreira y A. H. Tomita muestran en [20] la existencia de un espacio secuencialmente pseudocompacto que no es secuencialmente compacto. P. Lipparini demuestra en [25] que la propiedad (P) es equivalente a la siguiente propiedad en la clase de espacios topológicos: (P 0 ) Para toda sucesión hU n i de conjuntos abiertos no vacíos de X, existe un subconjunto infinito J ⇢ ! y un punto p 2 X tal que el conjunto {n 2 J : W \ U n = ;} es finito para toda vecindad W de p.…”