From 46634cd218b88705374f9b5dca09823a8ff0b194 Mon Sep 17 00:00:00 2001 From: Preston Pan Date: Thu, 19 Mar 2026 03:31:16 -0700 Subject: final commit before going to victoria --- mindmap/limit.org | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) (limited to 'mindmap') diff --git a/mindmap/limit.org b/mindmap/limit.org index f382f44..d25409b 100644 --- a/mindmap/limit.org +++ b/mindmap/limit.org @@ -90,7 +90,6 @@ In fact, this is sort of analogous to currying in the theory of computer science How do we know the space is compact? We know that $Y$ is compact because $[0, 1]$ is compact, and we apply Tychonoff's theorem. How do we know that $\overline{\phi(X)}$ is compact? It is closed and a subset of a compact set. However, what we have /not/ shown thus far is that $\phi(X)$ is truly an embedding. To see this, the completely regular property of $X$ saves the day; if we /didn't/ have this property, then it would be possible for some two points to /never/ be -separated by any function, and then you'd lose the one-to-one property of $\phi$. -\begin{align} -\lim x_{\alpha} -\end{align} +separated by any function, and then you'd lose the one-to-one property of $\phi$. Also, $\phi$ is clearly always continuous; we use the property that +$\pi_{\alpha}\circ \phi(x) = f_{\alpha}(x)$, and $\phi$ is continuous iff its projections $f_{\alpha}$ are continuous. Now all we need to show is that $\phi^{-1}$ is continuous, which we can +also do with the completely regular property. -- cgit v1.3