# Definition:System of Neighborhoods

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## Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $a \in A$.

Let $\mathcal N_a$ be the set of all neighborhoods of $a$ in $M$.

Then $\mathcal N_a$ is the **system of neighborhoods** of the point $a$.

## Also known as

The **system of neighborhoods** of a point can also be referred to as the **complete system of neighborhoods** of that point.

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.4$: Open Balls and Neighborhoods: Definition $4.4$