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

Taras Banakh; Dusan Repovs

doi:10.1016/j.topol.2010.01.010

paper

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

arXiv:0908.2228 · doi:10.1016/j.topol.2010.01.010

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 $u - l im X_{n}$ of the sequence $(X_{n})$ in the category of uniform spaces. This description implies that a function $f : u - l im X_{n} \to Y$ 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.

10 pages

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

Abstract

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