Filter on Set is Proper Filter

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Theorem

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ denote the power set of $S$.

Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the poset defined on $\mathcal P \left({S}\right)$ by the subset relation.


Let $\mathcal F$ be a filter on $S$.


Then $\mathcal F$ is a proper filter on $\left({\mathcal P \left({S}\right), \subseteq}\right)$.


Proof

From the general definition of a filter, we have:

A filter on $\left({S, \preccurlyeq}\right)$ is a subset $\mathcal F \subseteq S$ which satisfies the following conditions:

$\mathcal F \ne \varnothing$
$x, y \in \mathcal F \implies \exists z \in \mathcal F: z \preccurlyeq x, z \preccurlyeq y$
$\forall x \in \mathcal F: \forall y \in S: x \preccurlyeq y \implies y \in \mathcal F$

A filter $\mathcal F$ is proper if it does not equal $S$ itself.


From the definition of a filter on a set, we have:

A filter on $T$ is a set $\mathcal F \subset \mathcal P \left({T}\right)$ which satisfies the following conditions:

$T \in \mathcal F$
$\varnothing \notin \mathcal F$
$U, V \in \mathcal F \implies U \cap V \in \mathcal F$
$\forall U \in \mathcal F: U \subseteq V \subseteq T \implies V \in \mathcal F$


We can identify:

$\mathcal P \left({T}\right)$ with $S$
$\subseteq$ with $\preccurlyeq$.


Filter Not Empty

We have that $T \in \mathcal F$ and so $\mathcal F \ne \varnothing$.


Preceding Elements in Filter

We have that:

$U, V \in \mathcal F \implies U \cap V \in \mathcal F$

From Intersection is Subset, we have that $U \cap V \subseteq U$ and $U \cap V \subseteq V$.


So identifying $U$ with $x$, $V$ with $y$ and $U \cap V$ with $z$ it is clear that:

$x, y \in \mathcal F \implies \exists z \in \mathcal F: z \preccurlyeq x, z \preccurlyeq y$


Succeeding Elements in Filter

We have that:

$\forall U \in \mathcal F: U \subseteq V \subseteq T \implies V \in \mathcal F$

This can be rewritten:

$\forall U \in \mathcal F, V \in \mathcal P \left({T}\right): U \subseteq V \implies V \in \mathcal F$

Identifying $U$ with $x$ and $V$ with $y$, this translates as:

$\forall x \in \mathcal F, y \in S: x \preccurlyeq y \implies y \in \mathcal F$


Proper Filter

For $\mathcal F$ to be a proper filter on $\left({\mathcal P \left({T}\right), \subseteq}\right)$, it must not equal $\mathcal P \left({T}\right)$.

This is seen to be satisfied by the axiom $\varnothing \notin \mathcal F$.


All axioms are fulfilled, hence the result.

$\blacksquare$


Note about axioms

It seems at first glance that the demand $T \in \mathcal F$ is not axiomatic, as it is clear from the third property:

$U \in \mathcal F: U \subseteq T \subseteq T \implies T \in \mathcal F$

However, one of the properties of a filter is that it is specifically not empty.

Specifying that $T \in \mathcal F$ is therefore equivalent to specifying that $\mathcal F \ne \varnothing$.

Thus it would be possible to cite the first axiom as $\mathcal F \ne \varnothing$ instead, but this is rarely done.