Axiom:Neighborhood Space Axioms

From ProofWiki
Jump to navigation Jump to search


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 $\NN_x$      \(\ds \forall x \in S:\) \(\ds \NN_x \ne \O \)      
\((\text N 2)\)   $:$   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 $y \in N'$      \(\ds \forall x \in S: \forall N \in \NN_x:\) \(\ds \exists N' \in \NN_x, N' \subseteq N: \forall y \in N': N' \in \NN_y \)      

These stipulations are called the neighborhood space axioms.

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

Also see