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Typo fixes.

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This commit is contained in:
Bokuan Li
2026-03-15 12:32:31 -04:00
parent f951ccccdf
commit 62a8e78dfe
3 changed files with 29 additions and 18 deletions
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11
src/topology/uniform/metric.tex
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@@ -70,15 +70,16 @@ The axioms of uniform spaces strongly resembles working in a metric space. In fa
\begin{enumerate}
\item[(FB1)] For any $J, J' \subset I$ finite and $r, r' > 0$,
\[
\bigcap_{j \in J \cup J'}E(d_j, \min(r,r')E(d_j, r \wedge r') \subset \paren{\bigcap_{j \in J}E(d_j, r)} \cap \paren{\bigcap_{j \in J'}E(d_j, r')}
\bigcap_{j \in J \cup J'}E(d_j, r \wedge r') \subset \paren{\bigcap_{j \in J}E(d_j, r)} \cap \paren{\bigcap_{j \in J'}E(d_j, r')}
\]
\item[(UB1)] For each $i \in I$, $d(x, x) = 0$ for all $x \in X$. Thus for any $i \in I$ and $r > 0$, $E(d_i, r)$ contains the diagonal.
\item[(UB2)] For each $J \subset I$ finite and $r > 0$,
\[
\paren{\bigcap_{j \in J}E(d_j, r/2)} \circ \paren{\bigcap_{j \in J}E(d_j, r)} \subset \bigcap_{j \in J}E(d_j, r/2) \circ E(d_j, r/2) \subset \bigcap_{j \in J}E(d_j, r)
\]
\begin{align*}
\paren{\bigcap_{j \in J}E(d_j, r/2)} \circ \paren{\bigcap_{j \in J}E(d_j, r)} &\subset \bigcap_{j \in J}E(d_j, r/2) \circ E(d_j, r/2) \\
&\subset \bigcap_{j \in J}E(d_j, r)
\end{align*}
by the triangle inequality.
\end{enumerate}