# Definition:Neighborhood Space

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## Definition

Let $S$ be a set.

For each $x \in S$, let there be given a set $\mathcal N_x$ of subsets of $S$ which satisfy the neighborhood space axioms:

\((N1)\) | $:$ | There exists at least one element in $\mathcal N_x$ | \(\displaystyle \forall x \in S:\) | \(\displaystyle \mathcal N_x \ne \varnothing \) | ||||

\((N2)\) | $:$ | Each element $x$ is in its own $\mathcal N_x$ | \(\displaystyle \forall x \in S:\) | \(\displaystyle \forall N \in \mathcal N_x: x \in N \) | ||||

\((N3)\) | $:$ | Each superset of $N \in \mathcal N_x$ is also in $\mathcal N_x$ | \(\displaystyle \forall x \in S: \forall N \in \mathcal N_x:\) | \(\displaystyle N' \supseteq N \implies N' \in \mathcal N_x \) | ||||

\((N4)\) | $:$ | Intersection of $2$ elements of $\mathcal N_x$ is also in $\mathcal N_x$ | \(\displaystyle \forall x \in S: \forall M, N \in \mathcal N_x:\) | \(\displaystyle M \cap N \in N_x \) | ||||

\((N5)\) | $:$ | Exists $N' \subseteq N \in \mathcal N_x$ which is $\mathcal N_y$ of each $y \in N'$ | \(\displaystyle \forall x \in S: \forall N \in \mathcal N_x:\) | \(\displaystyle \exists N' \in \mathcal N_x, N' \subseteq N: \forall y \in N': N' \in \mathcal N_y \) |

The sets $\mathcal N_x$ are the neighborhoods of $x$ in $S$.

Let $\mathcal N$ be the set of open sets of $S$:

- $\mathcal N = \left\{{U \subseteq S: U}\right.$ is a neighborhood of each of its elements $\left.{}\right\}$

The set $S$ together with $\mathcal N$ is called a **neighborhood space** and is denoted $\left({S, \mathcal N}\right)$.

## Also see

- Results about
**neighborhood spaces**can be found here.

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 3.3$: Neighborhoods and Neighborhood Spaces: Definition $3.4$