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

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

2009-08-16

arxiv.org/abs/0908.2228v3

Topology Appl. 157:6 (2010), 1091-1100

Mathematics

General Topology

10 pages

Scientific paper

10.1016/j.topol.2010.01.010

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.

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