A note on local properties in products

TL;DR

This paper investigates conditions under which products of topological spaces preserve certain local properties, providing necessary and sufficient criteria for local properties like sequential compactness, Lindelöfness, and the Menger property.

Contribution

It establishes necessary and sufficient conditions for local properties to be preserved under finite products in topological spaces.

Findings

Characterization of when local properties are preserved under finite products

Examples include local sequential compactness, Lindelöfness, and the Menger property

Conditions are both necessary and sufficient for global property preservation

Abstract

We give conditions under which a product of topological spaces satisfies some local property. The conditions are necessary and sufficient when the corresponding global property is preserved under finite products. Further examples include local sequential compactness, local Lindel\"ofness, the local Menger property.