Definition:Neighborhood (Topology)/Point
Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $z \in S$ be a point in a $S$.
Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$.
Then $N_z$ is a neighborhood of $z$.
That is:
- $\exists U \in \tau: z \in U \subseteq N_z \subseteq S$
Neighborhood defined as Open
Some authorities define a neighborhood of a set $A$ as what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an open neighborhood:
- $N_A$ is a neighborhood of $A$ if and only if $N_A$ is an open set of $T$ which itself contains $A$.
That is, in order to be a neighborhood of $A$ in $T$, $N_A$ must not only be a subset of $T$, but also be an open set of $T$.
However, this treatment is less common, and considered by many to be old-fashioned.
When the term neighborhood is used on this site, it is assumed to be not necessarily open unless so specified.
Also see
- Results about neighborhoods can be found here.
Linguistic Note
The UK English spelling of neighborhood is neighbourhood.
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Definition $2.2$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.5$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): neighbourhood
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): neighbourhood: 2.