From 331401306cef5e3de0c11a29933e3056c8912c18 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 23 Sep 2026 15:48:24 -0400 Subject: [PATCH] Fixed up compactification. --- src/topology/main/compactify.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/topology/main/compactify.tex b/src/topology/main/compactify.tex index b331eeb..11d0393 100644 --- a/src/topology/main/compactify.tex +++ b/src/topology/main/compactify.tex @@ -105,7 +105,7 @@ In particular, $\bigcap_{A \in F}A^c \ne \emptyset$. As such, $\bracsn{A^c|A \in \mathcal{S}}$ generates an ultrafilter $\fU \in \beta X$. Since $\fU \not\in U_A$ for all $A \in \mathcal{S}$, $\mathcal{U}$ does not cover $X$, which is a contradiction. Thus $\beta X$ is compact. - Finally, since $X$ is discrete, $e$ is automatically continuous. For any $\emptyset \ne A \subset X$, $e(x) \in U_A$ for all $x \in A$, so $U_A \cap e(X) \ne \emptyset$, and $e(X)$ is dense in $\beta X$. Therefore $e(X)$ is embedded as a dense subspace of $\beta X$. + Finally, since $X$ is discrete, $e$ is automatically continuous. On the other hand, for each $x \in X$, $U_{\bracsn{x}} = e(\bracsn{x})$ is open in $\beta X$. In addition, for any $\emptyset \ne A \subset X$, $e(x) \in U_A$ for all $x \in A$, so $U_A \cap e(X) \ne \emptyset$, and $e(X)$ is dense in $\beta X$. Therefore $e(X)$ is embedded as a dense subspace of $\beta X$. (U1): Let $f \in C(X; [0, 1])$, and let $\beta f: \beta X \to [0, 1]$ be defined by $\beta f(\fU) = \lim_{\fU}f$. Fix $\fU \in \beta X$ and $\eps > 0$, then there exists $A \in \fU$ such that $f(A) \subset B_{[0, 1]}(\lim_{\fU}f, \eps)$. Thus for any $\fV \in U_A$, \[