Definition:Convergent Filter
Jump to navigation
Jump to search
Definition
Let $\struct {S, \tau}$ be a topological space.
Let $\FF$ be a filter on $S$.
Then $\FF$ converges to a point $x \in S$ if and only if:
- $\forall N_x \subseteq S: N_x \in \FF$
where $N_x$ is a neighborhood of $x$.
That is, a filter converges to a point $x$ if and only if every neighborhood of $x$ is an element of that filter.
If there is a point $x \in S$ such that $\FF$ converges to $x$, then $\FF$ is convergent.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Filters