Definition:Limit Point of Filter

Definition

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

A point $x \in S$ is called a limit point of $\mathcal F$ if:

$\displaystyle x \in \bigcap \left\{{\complement_S \left({V}\right) : V \in \mathcal F}\right\}$

where $\complement_S \left({V}\right)$ is the complement of $V$ relative to $S$.

$\mathcal F$ is said to converge on $x$.

Alternative Definition

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

A point $x \in X$ is called a limit point of $\mathcal F$ if $\mathcal F$ is finer than the neighborhood filter of $x$.

Also see

• Equivalent Definitions of Limit Point of Filter

• Results about filters can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Filters