Definition:Limit Point of Filter Base
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Definition
Let $\mathcal F$ be a filter on a set $S$.
Let $\mathcal B$ be a filter basis of $\mathcal F$.
A point $x \in S$ is called a limit point of $\mathcal B$ if $\mathcal F$ converges on $x$.
$\mathcal B$ is likewise said to converge on $x$.
Alternative Definition
A point $x \in S$ is called a limit point of $\mathcal B$ iff every neighborhood of $x$ contains a set of $\mathcal B$.
Explain what a neighborhood is in this context. As the definition currently lies, a neighborhood is defined in the context of a topology, while the filter itself is defined only on a basic, unadorned set.
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Also see
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Filters
