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 direct limits of sequences of uniform spaces in the categorical setting of uniform spaces and uniformly continuous maps. Let ((X_n){n\in\mathbb N}) be a sequence such that each inclusion (X_n\hookrightarrow X{n+1}) is a closed embedding. The authors construct the direct limit (u!-!\lim X_n) by endowing the set-theoretic union (\bigcup_n X_n) with the final uniformity generated by the family of entourages from the individual spaces. Concretely, for each (n) choose an entourage (U_n\in\mathcal U_{X_n}) and define
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