The topological structure of direct limits in the category of uniform spaces
Abstract
Let (Xn)n be a sequence of uniform spaces such that each space Xn is a closed subspace in Xn+1. We give an explicit description of the topology and uniformity of the direct limit u-lim Xn of the sequence (Xn) in the category of uniform spaces. This description implies that a function f:u-lim Xn Y to a uniform space Y is continuous if for every n the restriction f|Xn is continuous and regular at the subset Xn-1 in the sense that for any entourages U∈Y and V∈X there is an entourage V∈X such that for each point x∈ B(Xn-1,V) there is a point x'∈ Xn-1 with (x,x')∈ V and (f(x),f(x'))∈ U. Also we shall compare topologies of direct limits in various categories.
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