Definition:Basis for Neighborhood System
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Definition
Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$ be a point in $M$.
Let $\BB_a$ be a set of neighborhoods of $a$ in $M$.
Then $\BB_a$ is a basis for the neighborhood system at $a$ if and only if:
- $\forall N_a \subseteq M: \exists B \in \BB_a: B \subseteq N_a$
where $N_a$ denotes a neighborhood of $a$ in $M$.
That is, $\BB_a$ is a basis for the neighborhood system at $a$ if and only if every neighborhood of $a$ contains an element of $\BB_a$ as a subset.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Definition $4.9$