Initial \lambda-compactness in linearly ordered spaces

TL;DR

This paper characterizes initial -compactness in linearly ordered spaces, showing it is equivalent to -boundedness, and demonstrates that such properties are preserved under products.

Contribution

It provides a precise characterization of initial -compactness in linearly ordered spaces and proves the stability of this property under product operations.

Findings

Initial -compactness is equivalent to -boundedness in linearly ordered spaces.

Products of initially -compact linearly ordered spaces are also initially -compact.

The paper establishes a clear criterion linking topological and order-theoretic properties.

Abstract

We show that a linearly ordered topological space is initially \lambda-compact if and only if it is \lambda-bounded, that is, every set of cardinality $\leq \lambda$ has compact closure. As a consequence, every product of initially \lambda-compact linearly ordered topological spaces is initially \lambda-compact.