Topological structure of direct limits in the category of uniform spaces

Let ( X n ) n ∈ ω be a sequence of uniform spaces such that each space X n is a 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 → Y to a uniform space Y is continuous if for every n ∈ N the restriction f | X n is continuous and regular at the subset X n − 1 in the sense that for any entourages U ∈ U Y and V ∈ U X there is an entourage W ∈ U X such that for each point x ∈ B ( X n − 1 , W ) there is a point x ′ ∈ X n − 1 with ( x , x ′ ) ∈ V and ( f ( x ) , f ( x ′ ) ) ∈ U . Also we shall compare topologies of direct limits in various categories.