# Intersection of Neighborhoods in Topological Space is Neighborhood

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## Contents

## Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $x \in S$.

Let $M, N$ be a neighborhoods of $x$ in $T$.

Then $M \cap N$ is a neighborhood of $x$ in $T$.

That is:

- $\forall x \in S: \forall M, N \in \mathcal N_x: M \cap N \in N_x$

where $\mathcal N_x$ is the neighborhood filter of $x$.

## Proof

By definition of neighborhood:

- $\exists U_1 \in \tau: x \in U_1 \subseteq M$

where $U_1$ is an open set of $T$.

- $\exists U_2 \in \tau: x \in U_2 \subseteq N$

where $U_2$ is an open set of $T$.

Thus by Set Intersection Preserves Subsets:

- $U \subseteq M \cap N$

where $U = U_1 \cap U_2$

The result follows by definition of neighborhood of $a$.

$\blacksquare$

## Also see

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 3.3$: Neighborhoods and Neighborhood Spaces: Theorem $3.1: \ N 4$