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Diffstat (limited to 'mindmap/limit.org')
| -rw-r--r-- | mindmap/limit.org | 7 |
1 files changed, 3 insertions, 4 deletions
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. |