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

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.

💡 Research Summary

The paper investigates the topological and uniform structure of direct limits of uniform spaces. Given a tower of uniform spaces ((X_n){n\in\omega}) where each (X_n) is a closed subspace of (X{n+1}), the authors provide an explicit description of the topology and uniformity of the direct limit (u!-!\lim X_n) in the category of uniform spaces.
The main results are as follows:

1. Explicit Topology (Theorem 1.1). For any point (x) in the union (X=\bigcup_n X_n) and any sequence of entourages ((U_i){i\ge|x|}) with (U_i\in\mathcal U{X_i}), the set
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