Convergent sequences in various topological spaces

TL;DR

This paper explores the existence of non-trivial convergent sequences in various topological spaces, providing new proofs and insights related to Efimov's problem and the properties of the space βω.

Contribution

It introduces new proofs for the presence or absence of convergent sequences in specific classes of topological spaces, including ordered, scattered, metrisable spaces, and Valdivia compacta.

Findings

Ordered, scattered, and metrisable spaces contain convergent sequences.

The space βω has no non-trivial convergent sequences.

Cardinal coefficients influence the existence of convergent sequences.

Abstract

The following paper is inspired by Efimov's problem - an undecided problem of whether there exists an infinite compact topological space that does not contain neither non-trivial convergent sequences nor a copy of $\beta\omega$. After introducing basic topological concepts, we present several classes of topological spaces in which such sequences can certainly be found, namely ordered, scattered, metrisable spaces and Valdivia compacta. We show that some cardinal coefficients set limits on the smallest cardinality of the base and the smallest cardinality of a neighbourhood base, under which the existence of convergent sequences can be ensured. In the final part of the paper we define the space $\beta\omega$ and show its selected properties. In particular, we prove that there are indeed no non-trivial convergent sequences in $\beta\omega$. Whereas the statements of these theorems are commonly known, the proofs are notoriously difficult to find. In this paper we intend to fill that gap.