Tychonoff-like Product Theorems for Local Topological Properties

TL;DR

This paper establishes general criteria for when products of topological spaces possess local properties, linking local and global characteristics and extending classical theorems to broader classes of spaces.

Contribution

It provides a unified framework for characterizing local properties in product spaces using necessary and sufficient conditions, generalizing Tychonoff-like theorems.

Findings

Criteria for local properties in product spaces

Equivalence of local and global properties under certain conditions

Point-free formulations and decompositions into indecomposable factors

Abstract

We consider classes T of topological spaces (referred to as T-spaces) that are stable under continuous images and frequently under arbitrary products. A local T-space has for each point a neighborhood base consisting of subsets that are T-spaces in the induced topology. A general necessary and sufficient criterion for a product of topological spaces to be a local T-space in terms of conditions on the factors enables one to establish a broad variety of theorems saying that a product of spaces has a certain local property (like local compactness, local sequential compactness, local \sigma-compactness, local connectedness etc.) if and only if each factor has that local property, almost all have the corresponding global property, and not too many factors fail a suitable additional condition. Many of the results admit a point-free formulation; a look at sum decompositions into components of spaces with local properties yields product decompositions into indecomposable factors for certain classes of frames like completely distributive lattices or hypercontinuous frames.