topology on Solarized Sublime Sekai
https://vincenttam.gitlab.io/tags/topology/
Recent content in topology on Solarized Sublime SekaiHugo -- gohugo.iosere@live.hk (Vincent Tam)sere@live.hk (Vincent Tam)Fri, 05 Oct 2018 13:19:36 +0200Ultrafilters Are Maximal
https://vincenttam.gitlab.io/post/2018-10-05-ultrafilters-are-maximal/
Fri, 05 Oct 2018 13:19:36 +0200sere@live.hk (Vincent Tam)https://vincenttam.gitlab.io/post/2018-10-05-ultrafilters-are-maximal/Ultra filter : A filer $\mathcal{F}$ containing either $Y$ or $Y^\complement$ for any $Y \subseteq X$.
Two days ago, I spent an afternoon to understand Dudley's proof of this little result.
A filter is contained in some ultrafilter. A filter is an ultrafilter iff it's maximal.
At the first glance, I didn't even understand the organisation of the proof! I'm going to rephrase it for future reference.
only if: let $\mathcal{F}$ be an ultrafilter contained in another filter $\mathcal{G}$.Filters and Nets
https://vincenttam.gitlab.io/post/2018-10-03-filters-and-nets/
Wed, 03 Oct 2018 11:27:55 +0200sere@live.hk (Vincent Tam)https://vincenttam.gitlab.io/post/2018-10-03-filters-and-nets/Motivation \[ \gdef\vois#1#2{\mathcal{V}_{#1}(#2)} \]
Nets and filters are used for describing convergence in a non-metric space $X$.
Denote the collection of (open) neighbourhoods of $x \in X$ by \(\vois{X}{x}\).
Definitions and examples Directed set : A partially ordered set $I$ such that \(\forall i, j \in I: i \le j, \exists k \in I: k \ge j.\)
Net : A function in $X^I$, where $I$ is a directed set.
- example: any sequence in $X^\N$ Convergence of nets to a point : $x_i \to x$ if \(\forall A \in \vois{X}{x}, \exists j \in I: \forall k \ge j, x_k \in A.