In
mathematics, a
limit point of a
set S in a
topological space X is a point
x (which is in
X, but not necessarily in
S) that can be "approximated" by points of
S in the sense that every
neighbourhood of
x with respect to the
topology on
X also contains a point of
S other than
x itself. Note that x does not have to be an element of S. This concept profitably generalizes the notion of a
limit and is the underpinning of concepts such as
closed set and
topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.