Neighborhood
The neighborhood of a point is a notion which has slightly different meanings in different contexts.
General topology
Let be a topology, and let
be an element of
. We say that a set
is a neighborhood of
if there exists some open set
for which
.
Metric spaces
Let be a metric space, and let
be an element of
. A neighborhood
of
is the set of points
in
such that
, for some positive real
specific to
. The real
is called the radius of
. This neighborhood is sometimes denoted
. In metric spaces, neighborhoods are also called open balls.
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