Used "separated" instead of Hausdorff in the context of topological vector spaces.

src/fa/tvs/completion.tex

@@ -5,7 +5,7 @@
 \label{definition:tvs-completion}
     Let $E$ be a TVS over $K \in \RC$, then there exists $(\wh E, \iota)$ such that:
     \begin{enumerate}
-        \item $\wh E$ is a complete Hausdorff TVS.
+        \item $\wh E$ is a complete separated TVS.
         \item $\iota \in L(E; \wh E)$.
         \item[(U)] For any $(F, T)$ satisfying (1) and (2), there exists a unique $\ol{T} \in L(\wh E; F)$ such that the following diagram commutes:
     \end{enumerate}

src/fa/tvs/continuous.tex

@@ -54,7 +54,7 @@
 
 \begin{theorem}[Linear Extension Theorem (TVS)]
 \label{theorem:linear-extension-theorem-tvs}
-    Let $E$ be a TVS over $K \in \RC$, $F$ be a complete Hausdorff TVS over $K$, $D \subset E$ be a dense subspace, and $T \in L(D; F)$, then:
+    Let $E$ be a TVS over $K \in \RC$, $F$ be a complete separated TVS over $K$, $D \subset E$ be a dense subspace, and $T \in L(D; F)$, then:
     \begin{enumerate}
         \item There exists an extension $\ol T \in L(E; F)$ such that $\ol T|_D = T$.
         \item[(U)] For any $S \in C(E; F)$ satisfying (1), $S = \ol T$.

src/fa/tvs/quotient.tex

@@ -40,7 +40,7 @@
 
 \begin{proposition}
 \label{proposition:tvs-quotient-hausdorff}
-    Let $E$ be a TVS over $K \in \RC$, $M \subset E$ be a subspace, then $E/M$ is Hausdorff if and only if $M$ is closed.
+    Let $E$ be a TVS over $K \in \RC$, $M \subset E$ be a subspace, then $E/M$ is separated if and only if $M$ is closed.
 \end{proposition}
 \begin{proof}
     The space $M$ is closed if and only if
@@ -48,5 +48,5 @@
     M = \bigcap_{V \in \cn(0)}M + V
     \]
 
-    which is equivalent to $E/M$ being Hausdorff.
+    which is equivalent to $E/M$ being separated.
 \end{proof}