# 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 $\mathcal N(A).$

### Notes

• $X$ is a neighborhood of all of its points.
• Any open set $O\in\mathcal O$ is a neighborhood of all of its points.
• A neighborhood itself does not have to be open (e.g. it is closed but contains an open subset containing $A$, or it is neither open nor closed but contains a subset containing $A$).

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

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### References

#### Bibliography

1. Forster Otto: "Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984
2. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
3. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition