Simultaneous Sequential Compactness

TL;DR

This paper explores the concept of simultaneous convergence of sequences in various compact spaces, establishing bounds related to cardinal invariants and extending results to Hausdorff spaces.

Contribution

It introduces bounds on the number of sequences that can converge simultaneously in compact and Hausdorff spaces, linking these bounds to cardinal invariants like  and h.

Findings

Simultaneous convergence occurs for fewer than  sequences in spaces with weight .

In general spaces, simultaneous convergence occurs for fewer than h sequences.

The results are extended and their optimality is analyzed.

Abstract

A set of sequences is said to converge simultaneously if there exists an infinite subset $H$ of the index set $\omega$ such that all sequences converge when restricted to $H$. We discuss simultaneous convergence of sequences in the same or in different sequentially compact spaces; we link the results for different spaces to ones for the same space; we show that simultaneous convergence happens for less than $\mathfrak s$ sequences in spaces with weight bounded by $\mathfrak s$ and for less than $\mathfrak h$ sequences in general; we show a slight generalisation of these results in the context of Hausdorff spaces; and finally we investigate their optimality.