# The topological structure of direct limits in the category of uniform spaces - Mathematics > General Topology

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Abstract: Let $X n {n}$ be a sequence of uniform spaces such that each space $X n$ isa closed subspace in $X {n+1}$. We give an explicit description of the topologyand uniformity of the direct limit $u-lim X n$ of the sequence $X n$ in thecategory of uniform spaces. This description implies that a function $f:u-limX n\to Y$ 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 forany entourages $U\in\U Y$ and $V\in\U X$ there is an entourage $V\in\U X$ suchthat for each point $x\in BX {n-1},V$ there is a point $x-\in X {n-1}$ with$x,x-\in V$ and $fx,fx-\in U$. Also we shall compare topologies ofdirect limits in various categories.

Author: ** Taras Banakh, Dusan Repovs**

Source: https://arxiv.org/