Definition
If is a topological space and is a point in, a neighbourhood of is a subset of, which includes an open set containing ,
This is also equivalent to being in the interior of .
Note that the neighbourhood need not be an open set itself. If is open it is called an open neighbourhood. Some authors require that neighbourhoods be open, so it is important to note conventions.
A set which is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points.
The collection of all neighbourhoods of a point is called the neighbourhood system at the point.
If is a subset of then a neighbourhood of is a set which includes an open set containing . It follows that a set is a neighbourhood of if and only if it is a neighbourhood of all the points in . Furthermore, it follows that is a neighbourhood of iff is a subset of the interior of .
Read more about this topic: Neighbourhood (mathematics)
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