The 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 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 ∈ 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 V ∈ UX 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. Introduction In this paper we shall give an explicit description of the topology and the uniformity of direct limits of sequences of uniform spaces in the category of uniform spaces. The obtained results will be essentially in the topological characterization of LF-spaces, see [Ba2]. By definition, an LF-space is the direct limit lc-lim −→n of a tower X0 ⊂ X1 ⊂ X2 ⊂ · · · of Fréchet (= locally convex complete metric linear) spaces in the category in locally convex spaces. Thus, lc-lim −→n is the linear space X = ⋃ n∈ω Xn endowed with the strongest topology that turns X into a locally convex linear topological space such that the identity maps Xn → X are continuous. The union X = ⋃ n∈ω Xn endowed with the strongest topology making the identity maps Xn → X , n ∈ ω, continuous is called the topological direct limit of the tower (Xn) and is denoted by t-lim −→n It follows from the definitions that the identity map t-lim −→n → lc-lim −→n is continuous. In general, this map is not a homeomorphism, which means that the topology of topological direct limit t-lim −→n can be strictly larger than the topology of the locally convex direct limit lc-lim −→n Between the topologies of topological and locally convex direct limits there is a spectrum of direct limit topologies in categories that are intermediate between the category of topological and locally convex spaces. The most important examples of such categories are the categories of linear topological spaces, of topological groups and the category of uniform spaces. The direct limits of towers (Xn) in those categories will be denoted by l-lim −→n g-lim −→n and u-lim −→n respectively. The direct limit g-lim −→n (resp. l-lim −→n of a tower (Xn)n∈ωof topological groups (resp. linear topological spaces) is the union X = 1991 Mathematics Subject Classification. 46A13; 54B30; 54E15; 54H11.