Axiom:Neighborhood Space Axioms

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Axioms

A neighborhood space is a set $S$ such that, for each $x \in S$, there exists a set of subsets $\NN_x$ of $S$ satisfying the following conditions:

\((\text N 1)\)   $:$   There exists at least one element in $\NN_x$      \(\displaystyle \forall x \in S:\) \(\displaystyle \NN_x \ne \O \)             
\((\text N 2)\)   $:$   Each element $x$ is in its own $\NN_x$      \(\displaystyle \forall x \in S:\) \(\displaystyle \forall N \in \NN_x: x \in N \)             
\((\text N 3)\)   $:$   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 $y \in N'$      \(\displaystyle \forall x \in S: \forall N \in \NN_x:\) \(\displaystyle \exists N' \in \N N_x, N' \subseteq N: \forall y \in N': N' \in \NN_y \)             

These stipulations are called the neighborhood space axioms.


Each element of $\NN_x$ is called a neighborhood of $x$.


Also see


Sources