Definition: Neighborhood

Let $A$ be an element of a topological space $(X,\mathcal O)$ ($A$ might be a single point of a set of points). A neighborhood $N_A$ is any subset of $X$ containing an open set containing $A$, formally $$N_A\in \{U\subseteq X\mid \exists O\in\mathcal O, A\in O\subseteq U\}.$$

The set of all neighborhoods of $A$ is denoted by $\mathcal N(A).$


Axioms: 1
Definitions: 2
Parts: 3
Proofs: 4 5 6 7 8
Propositions: 9 10 11

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  3. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition