Topology MOC

Neighbourhood

In topology, a neighbourhood¹ of a point is a set containing some open set around a point. topology An open neighbourhood of a point is then an open set containing a point. Some authors use a different definition where all neighbourhoods are open,² but this will be distinguished in these notes (see Topology notation in these notes)

It follows from this definition that a set is open iff it is a neighbourhood of all its points.

• Neighbourhood system
• Neighbourhood basis

Examples

In a metric space

Let (𝑋,𝑑) be a metric space, and 𝑥 ∈𝑋 Then 𝑆 ⊆𝑋 is said to be a neighbourhood of 𝑥 iff there exists 𝜖 >0 such that

𝑥∈𝐵𝜖(𝑥)⊆𝑆⊆𝑋

which follows if 𝐵𝜖(𝑥) ⊆𝑆.

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Footnotes

1. German Umgebung von 𝑥 ↩

2. 2000, Topology, pp. 96–97 ↩