The topological structure of direct limits in the category of uniform spaces

TL;DR

This paper explicitly describes the topology and uniformity of direct limits of sequences of uniform spaces, providing criteria for continuity of functions and comparing topologies across categories.

Contribution

It offers a detailed description of the topology and uniformity of direct limits in the category of uniform spaces, including continuity criteria and comparisons across categories.

Findings

Explicit topology and uniformity description for direct limits.

Continuity characterized by restrictions and regularity conditions.

Comparison of direct limit topologies in various categories.

Abstract

Let $(X_n)_{n}$ be a sequence of uniform spaces such that each space $X_n$ is a closed subspace in $X_{n+1}$. We give an explicit description of the topology and uniformity of the direct limit $u-lim X_n$ of the sequence $(X_n)$ in the category of uniform spaces. This description implies that a function $f:u-lim X_n\to Y$ to a uniform space $Y$ is continuous if for every $n$ the restriction $f|X_n$ is continuous and regular at the subset $X_{n-1}$ in the sense that for any entourages $U\in\U_Y$ and $V\in\U_X$ there is an entourage $V\in\U_X$ such that for each point $x\in B(X_{n-1},V)$ there is a point $x'\in X_{n-1}$ with $(x,x')\in V$ and $(f(x),f(x'))\in U$. Also we shall compare topologies of direct limits in various categories.