Category:Neighborhood Spaces

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This category contains results about Neighborhood Spaces.

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 $\NN_x$      \(\ds \forall x \in S:\) \(\ds \NN_x \ne \O \)      
\((\text N 2)\)   $:$   Each element of $\NN_x$ contains $x$      \(\ds \forall x \in S:\) \(\ds \forall N \in \NN_x: x \in N \)      
\((\text N 3)\)   $:$   Each superset of $N \in \NN_x$ is also in $\NN_x$      \(\ds \forall x \in S: \forall N \in \NN_x:\) \(\ds N' \supseteq N \implies N' \in \NN_x \)      
\((\text N 4)\)   $:$   The intersection of $2$ elements of $\NN_x$ is also in $\NN_x$      \(\ds \forall x \in S: \forall M, N \in \NN_x:\) \(\ds M \cap N \in N_x \)      
\((\text N 5)\)   $:$   There exists $N' \subseteq N \in \NN_x$ which is $\NN_y$ of each $y \in N'$      \(\ds \forall x \in S: \forall N \in \NN_x:\) \(\ds \exists N' \in \NN_x, N' \subseteq N: \forall y \in N': N' \in \NN_y \)      

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 $\struct {S, \NN}$.