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Fixed typo.
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Bokuan Li
2026-06-30 19:34:11 -04:00
parent 1f9e0bea78
commit 89ec2234be
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src/topology/main/stonean.tex
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@@ -9,7 +9,7 @@
\begin{theorem}[Stone-Nakano] \begin{theorem}[Stone-Nakano]
\label{theorem:stone-nakano-extremely-disconnected} \label{theorem:stone-nakano-extremely-disconnected}
Let $X$ be a topological space. If $X$ is extremely disconnected, then $C(X; \real)$ is order complete. Conversely, if $X$ is ??? and $C(X; \real)$ is order complete, then $X$ is extremely disconnected. Let $X$ be a topological space. If $X$ is extremely disconnected, then $C(X; \real)$ is order complete. Conversely, if $X$ is completely regular and $C(X; \real)$ is order complete, then $X$ is extremely disconnected.
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
($\Rightarrow$): Suppose that $X$ is extremely disconnected. Let $\cf \subset C(X; \real)$ and $F \in C(X; \real)$ be an upper bound of $\cf$. For each $q \in \real$, let ($\Rightarrow$): Suppose that $X$ is extremely disconnected. Let $\cf \subset C(X; \real)$ and $F \in C(X; \real)$ be an upper bound of $\cf$. For each $q \in \real$, let