# Definition:Neighborhood Space

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## Definition

Let $S$ be a set.

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

\((\text N 1)\) | $:$ | There exists at least one element in $\NN_x$ | \(\displaystyle \forall x \in S:\) | \(\displaystyle \NN_x \ne \O \) | ||||

\((\text N 2)\) | $:$ | Each element of $\NN_x$ contains $x$ | \(\displaystyle \forall x \in S:\) | \(\displaystyle \forall N \in \NN_x: x \in N \) | ||||

\((\text N 3)\) | $:$ | Each superset of $N \in \NN_x$ is also in $\NN_x$ | \(\displaystyle \forall x \in S: \forall N \in \NN_x:\) | \(\displaystyle N' \supseteq N \implies N' \in \NN_x \) | ||||

\((\text N 4)\) | $:$ | The intersection of $2$ elements of $\NN_x$ is also in $\NN_x$ | \(\displaystyle \forall x \in S: \forall M, N \in \NN_x:\) | \(\displaystyle M \cap N \in N_x \) | ||||

\((\text N 5)\) | $:$ | There exists $N' \subseteq N \in \NN_x$ which is $\NN_y$ of each $y \in N'$ | \(\displaystyle \forall x \in S: \forall N \in \NN_x:\) | \(\displaystyle \exists N' \in \NN_x, N' \subseteq N: \forall y \in N': N' \in \NN_y \) |

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

Let $\NN$ be the set of open sets of $S$:

- $\NN = \leftset {U \subseteq S: U}$ is a neighborhood of each of its elements$\rightset {}$

The set $S$ together with $\NN$ is called a **neighborhood space** and is denoted $\struct {S, \NN}$.

## Also see

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

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Definition $3.4$