Definition:Convergent Filter Basis
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Definition
Let $\struct {S, \tau}$ be a topological space.
Let $\BB$ be a filter basis of a filter $\FF$ on $S$.
Then $\BB$ converges to a point $x \in S$ if and only if:
- $\forall N_x \subseteq S: \exists B \in \BB: B \subseteq N_x$
where $N_x$ is a neighborhood of $x$.
That is, a filter basis is convergent to a point $x$ if every neighborhood of $x$ contains some set of that filter basis.
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