Added projective limits for LC spaces.

src/topology/uniform/uc.tex

@@ -44,6 +44,7 @@
         \]
 
         is a fundamental system of entourages for $\fU$.
+        \item[(4)] For any uniform space $Y$ and map $f: Y \to X$, $f \in UC(Y; X)$ if and only if $f_i \circ f \in UC(Y; Y_i)$ for all $i \in I$.
     \end{enumerate}
 
 
@@ -59,6 +60,12 @@
     (1): $\fU \supset (f_i \times f_i)^{-1}(\fU_i)$ for all $i \in I$.
 
     (U): For any $i \in I$, $\mathfrak{V} \supset (f_i \times f_i)^{-1}(\fU_i)$. By (F2), $\mathfrak{V} \supset \fB$, so $\mathfrak{V} \supset \fU$.
+
+    (4): Let $J \subset I$ finite and $\seqj{U_j}$ such that $U_j \in \fU_j$ for each $j \in J$, then
+    \[
+    (f \times f)^{-1}\paren{\bigcap_{j \in J}(f_j \times f_j)^{-1}(U_j)} = \bigcap_{j \in J}[(f_j \circ f) \times (f_j \circ f)]^{-1}(U_j)
+    \]
+    is an entourage of $Y$.
 \end{proof}