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4.3 Filters

4.3.1 Definition of a filter.

Let \(X\) be a set. A filter on \(X\) is a family \(\sf \) of subsets of \(X\) such that:

  • 1. \(\sf \neq \emp \) and \(\emp \nin \sf \);

  • 2. if \(F\in \sf \) and \(F\sub F’\sub X\), then \(F’\in \sf \);

  • 3. if \(F_{1},F_{2}\in \sf \), then \(F_{1}\cap F_{2}\in \sf \).

Examples.

If \(X\) is any nonempty set and \(\sf :=\set X\), then \(\sf \) is a filter on \(X\).

Let \(X\) be an infinite set and \(\sf \) consist of all cofinite sets. Then \(\sf \) is a filter on \(X\).

4.3.2 Filter generated by a net.

Let \(\net x{\ga }A\) be a net in a set \(X\) and \(\sf \) consist of those \(F\sub \sf \) for which there exist \(\gb \in A\) such that \(x_{\ga }\in F\) for every \(\ga \ge \gb \), that is \(F\in \sf \) provided the net is eventually in \(F\). Then \(\sf \) is a filter on \(X\).

4.3.3 Nbhd filter.

Let \(X\) be a topological space, \(x\in X\) and \(\sf \) consist of all nbhds of \(x\). Then \(\sf \) is a filter. It is called the nbhd filter at \(x\).

4.3.4 Comparing filters.

Let \(\sf \) and \(\sf ’\) be filters on a set \(X\). Then \(\sf \) is finer than \(\sf ’\) when \(\sf ’\sub \sf \).

4.3.5 Convergence of filters.

Let \(\sf \) be a filter in a topological space \(X\). Then \(\sf \) converges to \(x\in X\) provided \(\sf \) is finer than the nbhd filter at \(x\).

4.3.6 Theorem.

Let \(X\) be a topological space, \(\net x{\ga }A\) be a net in \(X\) and \(\sf \) be the filter generated by \(\net x{\ga }A\). Then, for every \(x\in X\), the net \(\net x{\ga }A\) converges to \(x\) if and only if \(\sf \) converges to \(x\) .

  • Proof. (to be written)  □

4.3.7 Ultrafilters.

A filter \(\sf \) on a set \(X\) is an ultrafilter provided every filter on \(X\) that is finer than \(\sf \) must be equal \(\sf \).

4.3.8 Theorem.

A filter \(\sf \) on a set \(X\) is an ultrafilter if and only if for every \(Y\sub X\) either \(Y\in \sf \) or \(X\sem Y\in \sf \).

  • Proof. (to be written)  □

4.3.9 Theorem.

Let \(\net x{\ga }A\) be a net in a set \(X\) and \(\sf \) be the filter generated by \(\net x{\ga }A\). Then \(\sf \) is an ultrafilter if and only if \(\net x{\ga }A\) is an ultranet.

  • Proof. (to be written)  □

4.3.10 Zorn’s Lemma.

If \(X\) is a partially ordered set and each chain in \(X\) has an upper bound, then there exists a maximal element in \(X\).

4.3.11 Theorem.

Let \(X\) be a set and \(\sf \) be a filter on \(X\). Then there exists an ultrafilter on \(X\) that is finer than \(\sf \).

  • Proof. (to be written)  □

4.3.12 Theorem.

Let \(X\) be a set and \(\net x{\ga }A\) be a net in \(X\). Then there exists a subnet \(\net y{\gb }B\) of \(\net x{\ga }A\) that is an ultranet.

  • Proof. Let \(\sf \) be the filter on \(X\) that is generated by \(\net x{\ga }A\) and \(\su \) be an ultrafilter on \(X\) that is finer than \(\sf \). Define

    \[ B:=\set {\ang {\ga ,U}:\ga \in A,\,U\in \sf ,\,x_{\ga }\in U} \]

    and let \(B\) be directed by

    \[ \ang {\ga ,U}\le \ang {\ga ’,U’} \]

    if \(\ga \le \ga ’\) and \(U’\sub U\). If \(\ph :B\to A\) is given by \(\ph \of {\ga ,U}:=\ga \), then \(\ph \) defines a subnet \(\net y{\gb }B\) of \(\net x{\ga }A\). We show that \(\net y{\gb }B\) is an ultranet.

    Let \(Y\sub X\). Then either \(Y\) or \(X\sem Y\) belongs to \(\su \). If \(Y\in \su \), then \(X\sem Y\nin \su \), which implies that \(X\sem Y\nin \sf \). Thus the net \(\net x{\ga }A\) is not eventually in \(X\sem Y\) and hence \(\net x{\ga }A\) is frequently in \(Y\). Let \(\ga _{0}\in A\) be such that \(x_{\ga _{0}}\in Y\). Then

    \[ \gb _{0}:=\ang {\ga _{0},Y}\in B. \]

    If \(\gb :=\ang {\ga ,U}\ge \gb _{0}\), then \(x_{\ga }\in U\sub Y\) so \(y_{\gb }=x_{\ga }\in Y\). Thus \(\net y{\gb }B\) is eventually in \(Y\). If \(X\sem Y\in \su \), then a similar argument shows that \(\net y{\gb }B\) is eventually in \(X\sem Y\).  □