Definition:Neighborhood Space

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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.