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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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Related concepts (3)
Accumulation point
In mathematics, a limit point, accumulation point, or cluster point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences.
Adherent point
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
Closure (topology)
In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "very near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.
Related lectures (19)
Interior Points and Compact Sets
Explores interior points, boundaries, adherence, and compact sets, including definitions and examples.
Point Interior and Boundary in Real Numbers
Explores point interior and boundary concepts in real numbers, covering definitions and implications.
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Explores convergence of sequences in closed sets and the importance of understanding convergence in relation to closedness.