Products of straight spaces

TL;DR

This paper characterizes when products of metric spaces are straight, focusing on conditions like precompactness and local connectedness, and fully solves the problem for infinite products of uniformly locally connected spaces.

Contribution

It provides necessary and sufficient conditions for the straightness of finite and infinite products of metric spaces, extending understanding of product stability in metric space properties.

Findings

Product of two straight spaces is straight iff both are straight and meet specific conditions.

Finite products of ULC spaces are straight, but not all products of straight spaces are.

Infinite products of ULC spaces are characterized for straightness and ULC properties.

Abstract

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. A locally connected space is straight if it is uniformly locally connected (ULC). It is easily seen that ULC spaces are stable under finite products. On the other hand the product of two straight spaces is not necessarily straight. We prove that the product X x Y of two metric spaces is straight if and only if both X and Y are straight and one of the following conditions holds: (a) both X and Y are precompact; (b) both X and Y are locally connected; (c) one of the spaces is both precompact and locally connected. In particular, when X satisfies (c), the product X x Z is straight for every straight space Z. Finally, we characterize when infinite products of metric spaces are ULC and we completely solve the problem of straightness of infinite products of ULC spaces.