In mathematics, an adherent point (also closure point or point of closure or contact point)[1] of a subset
of a topological space
(or equivalently, every open neighborhood of
is an adherent point for
thus This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of
Thus every limit point is an adherent point, but the converse is not true.
is either a limit point of
Intuitively, having an open set
defined as the area within (but not including) some boundary, the adherent points of
including the boundary.
is a non-empty subset of
which is bounded above, then the supremum
is an adherent point that is not in the interval, with usual topology of
of a metric space
is (sequentially) closed in
is a topological subspace of
is endowed with the subspace topology induced on it by
will follow once it is shown that
(by definition of the subspace topology) so that
For the converse, assume that
will follow once it is shown that
By definition of the subspace topology, there exists a neighborhood
in every (or alternatively, in some) topological superspace of
is a subset of a topological space then the limit of a convergent sequence in
does not necessarily belong to
Then by definition of limit, for all neighbourhoods
In contrast to the previous example, the limit of a convergent sequence in
is not necessarily a limit point of
is the constant sequence
is not a limit point of