section we shall discuss the notion of the direct limit in the category of uniform spaces and their uniformly continuous maps. In Section \[s:ulim-top\] we shall apply those results to show that for a tower $(G_n)_{n\in{\omega}}$ of topological groups the topologies $\LA[\tau]$ and $\RA[\tau]$ on the union $G=\bigcup_{n\in{\omega}}G_n$ are generated by uniformities of direct limits of the groups $G_n$ endowed with the left and right uniformities.

Fundamenta of the theory of uniform spaces can be found in [@En Ch.8]. Uniformities on groups are thoroughly discussed in [@RD] and [@AT §1.8]. In the sequel, for a uniform space $X$ by ${\mathcal U}_X$ we shall denote the uniformity of $X$.

Let $$X_0\to X_1\to X_2\to\cdots$$be a sequence of uniform spaces and their injective uniformly continuous maps. We shall identify each space $X_n$ with a subset of the uniform space $X_{n+1}$, carrying its own uniformity, which is stronger than that inherited from $X_{n+1}$. By the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}X_n$ of the sequence of uniform spaces $(X_n)_{n\in{\omega}}$ we understand the union $X=\bigcup_{n\in{\omega}}X_n$ endowed with the strongest (not necessarily separated) uniformity turning the identity inclusions $X_n\to X$, $n\in{\omega}$, into uniformly continuous maps.

A sequence $$X_0\to X_1\to X_2\to\cdots$$ of uniform spaces is called a [*tower*]{} of 