In
mathematics, an adherent point (also closure point or point of closure or contact point)
[ Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15.] of a
subset of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is a point
in
such that every
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
of
(or equivalently, every
open neighborhood
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a p ...
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 mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
, 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
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
.
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
Empty set, 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
Topological space, topology of
A subset
of a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
contains all of its adherent points if and only if
is (
sequentially
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
)
closed in
Adherent points and subspaces
Suppose
and
where
is a
topological subspace
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
of
(that is,
is endowed with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
) 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
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
, 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
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
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
s
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
See also
*
*
*
*
*
Notes
Citations
References
* Adamson, Iain T.,
A General Topology Workbook', Birkhäuser Boston; 1st edition (November 29, 1995). .
*
Apostol, Tom M., ''Mathematical Analysis'', Addison Wesley Longman; second edition (1974).
*
Lipschutz, Seymour; ''Schaum's Outline of General Topology'', McGraw-Hill; 1st edition (June 1, 1968). .
*
L.A. Steen,
J.A.Seebach, Jr., ''Counterexamples in topology'', (1970) Holt, Rinehart and Winston, Inc..
*{{PlanetMath attribution, urlname=adherentpoint, title=Adherent point
General topology