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 $\mathcal{G}$. [Read More]

Filters and Nets

Some basic examples

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. [Read More]