# Selective sequential pseudocompactness

## TL;DR

This paper introduces the class of selectively sequentially pseudocompact (SSP) spaces, explores their properties, closure under various operations, and constructs examples of SSP group topologies, expanding understanding of generalized compactness.

## Contribution

It defines the SSP class, proves its closure properties, relates it to existing classes, and constructs new SSP group topologies with many generators.

## Key findings

- SSP spaces are closed under products and continuous images.
- All omega-bounded groups are SSP, but not all compact spaces are.
- Constructed SSP group topologies on free and free Abelian groups with continuum-many generators.

## Abstract

We say that a topological space X is selectively sequentially pseudocompact (SSP for short) if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in U_n for every n in such a way that the sequence (x_n) has a convergent subsequence. We show that the class of SSP spaces is closed under taking arbitrary products and continuous images, contains the class of all dyadic spaces and forms a proper subclass of the class of strongly pseudocompact spaces introduced recently by Garc\'ia-Ferreira and Ortiz-Castillo. We investigate basic properties of this new class and its relations with known compactness properties. We prove that every omega-bounded (=the closure of which countable set is compact) group is SSP, while compact spaces need not be SSP. Finally, we construct SSP group topologies on both the free group and the free Abelian group with continuum-many generators.