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Concept
Adherent point
Summary
In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset of a topological space is a point in such that every neighbourhood of (or equivalently, every open neighborhood of ) contains at least one point of A point is an adherent point for if and only if is in the closure of thus
if and only if for all open subsets if
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 contains at least one point of Thus every limit point is an adherent point, but the converse is not true. An adherent point of is either a limit point of or an element of (or both). An adherent point which is not a limit point is an isolated point.
Intuitively, having an open set defined as the area within (but not including) some boundary, the adherent points of are those of including the boundary.
If is a non-empty subset of which is bounded above, then the supremum is adherent to In the interval is an adherent point that is not in the interval, with usual topology of
A subset of a metric space contains all of its adherent points if and only if is (sequentially) closed in
Suppose and where is a topological subspace of (that is, is endowed with the subspace topology induced on it by ). Then is an adherent point of in if and only if is an adherent point of in
By assumption, and Assuming that let be a neighborhood of in so that will follow once it is shown that The set is a neighborhood of in (by definition of the subspace topology) so that implies that Thus as desired. For the converse, assume that and let be a neighborhood of in so that will follow once it is shown that By definition of the subspace topology, there exists a neighborhood of in such that Now implies that From it follows that and so as desired.
Consequently, is an adherent point of in if and only if this is true of in every (or alternatively, in some) topological superspace of
If is a subset of a topological space then the limit of a convergent sequence in does not necessarily belong to however it is always an adherent point of Let be such a sequence and let be its limit.
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