Topological structure of direct limits in the category of uniform spaces

Let (Xn)n∈ω be a sequence of uniform spaces such that each space Xn is a subspace in Xn+1. We give an explicit description of the topology and uniformity of the direct limit of the sequence (Xn) in the category of uniform spaces. This description implies that a function to a uniform space Y is continuous if for every n∈N the restriction f|Xn is continuous and regular at the subset Xn−1 in the sense that for any entourages U∈UY and V∈UX there is an entourage W∈UX such that for each point x∈B(Xn−1,W) 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.