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diff --git a/mindmap/limit.org b/mindmap/limit.org index 27531a2..f382f44 100644 --- a/mindmap/limit.org +++ b/mindmap/limit.org @@ -67,3 +67,30 @@ s_{0} \le ... \le s_{n} \le s_{n + 1} \le s_{n + 2} \le ... \le M \end{align} #+end_proof +** Limits as Objects +Limits can also be objects. This is most aptly demonstrated in more abstract fields of mathematics such as algebraic topology, +where the central "object of importance" (a common theme in math is one where you have an object of importance) is the net. +Specifically, the limits of universal nets have a deep relation to compactness, but here we will explore the most informative and essential +form of this idea and its algebraic properties. We will quickly go over the one-point compactification, and then introduce the stone-cech +compactification after. +*** One Point Compactification +:PROPERTIES: +:ID: 339b32e7-ad89-40d7-8b11-5b293bd1056f +:END: +*** Stone Cech Compcatification +:PROPERTIES: +:ID: 14bebb09-2e38-4b55-adc0-97ba571331af +:END: +We can construct the Stone Cech Compcatification on a completely regular topological space $X$, which will require a specific construction +but will at least give us the Hausdorff property in the compactified space. To start, let $A$ be the set of all $f_{\alpha}: X \rightarrow [0, 1]_{\alpha}$ such that $f$ is +continuous (with $\alpha$ being an arbitrary but consistent index), and let us define a Tychonoff space $Y = \prod_{\alpha \in A}[0, 1]_{\alpha}$ and an embedding $\phi: X \rightarrow Y$ +where the embedding $\phi$ is defined as $(\phi(x))_{\alpha }= f_{\alpha}(x)$. Then the idea is that the /closure/ of $\phi(X)$ in $Y$ is a compactification of $X$. +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} |