Definition:Neighborhood Space

From ProofWiki
Jump to navigation Jump to search


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.