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Point that belongs to the closure of some give subset of a topological space
This definition differs from that of a limit point, in that for a limit point it is required that every neighborhood of contains at least one point of different from 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.
Examples and sufficient conditions
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
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 [proof 1] Consequently, is an adherent point of in if and only if this is true of in every (or alternatively, in some) topological superspace of
Adherent points and sequences
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. Then by definition of limit, for all neighbourhoods of there exists such that for all In particular, and also so is an adherent point of
In contrast to the previous example, the limit of a convergent sequence in is not necessarily a limit point of ; for example consider as a subset of Then the only sequence in is the constant sequence whose limit is but is not a limit point of it is only an adherent point of
Closed set – The complement of an open subset a topological space. It contains all points that are "close" to it.
^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.