Neighborhood
From AoPSWiki
The neighborhood of a point is a notion which has slightly different meanings in different contexts. Informally, a neighborhood of
in some space
is a set that contains all points "sufficiently close" to
. This notion may be formalized differently depending on the nature of the space.
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.
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
.
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