- Source: Adherent point
In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset
A
{\displaystyle A}
of a topological space
X
,
{\displaystyle X,}
is a point
x
{\displaystyle x}
in
X
{\displaystyle X}
such that every neighbourhood of
x
{\displaystyle x}
(or equivalently, every open neighborhood of
x
{\displaystyle x}
) contains at least one point of
A
.
{\displaystyle A.}
A point
x
∈
X
{\displaystyle x\in X}
is an adherent point for
A
{\displaystyle A}
if and only if
x
{\displaystyle x}
is in the closure of
A
,
{\displaystyle A,}
thus
x
∈
Cl
X
A
{\displaystyle x\in \operatorname {Cl} _{X}A}
if and only if for all open subsets
U
⊆
X
,
{\displaystyle U\subseteq X,}
if
x
∈
U
then
U
∩
A
≠
∅
.
{\displaystyle x\in U{\text{ then }}U\cap A\neq \varnothing .}
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
x
{\displaystyle x}
contains at least one point of
A
{\displaystyle A}
different from
x
.
{\displaystyle x.}
Thus every limit point is an adherent point, but the converse is not true. An adherent point of
A
{\displaystyle A}
is either a limit point of
A
{\displaystyle A}
or an element of
A
{\displaystyle A}
(or both). An adherent point which is not a limit point is an isolated point.
Intuitively, having an open set
A
{\displaystyle A}
defined as the area within (but not including) some boundary, the adherent points of
A
{\displaystyle A}
are those of
A
{\displaystyle A}
including the boundary.
Examples and sufficient conditions
If
S
{\displaystyle S}
is a non-empty subset of
R
{\displaystyle \mathbb {R} }
which is bounded above, then the supremum
sup
S
{\displaystyle \sup S}
is adherent to
S
.
{\displaystyle S.}
In the interval
(
a
,
b
]
,
{\displaystyle (a,b],}
a
{\displaystyle a}
is an adherent point that is not in the interval, with usual topology of
R
.
{\displaystyle \mathbb {R} .}
A subset
S
{\displaystyle S}
of a metric space
M
{\displaystyle M}
contains all of its adherent points if and only if
S
{\displaystyle S}
is (sequentially) closed in
M
.
{\displaystyle M.}
= Adherent points and subspaces
=Suppose
x
∈
X
{\displaystyle x\in X}
and
S
⊆
X
⊆
Y
,
{\displaystyle S\subseteq X\subseteq Y,}
where
X
{\displaystyle X}
is a topological subspace of
Y
{\displaystyle Y}
(that is,
X
{\displaystyle X}
is endowed with the subspace topology induced on it by
Y
{\displaystyle Y}
). Then
x
{\displaystyle x}
is an adherent point of
S
{\displaystyle S}
in
X
{\displaystyle X}
if and only if
x
{\displaystyle x}
is an adherent point of
S
{\displaystyle S}
in
Y
.
{\displaystyle Y.}
Consequently,
x
{\displaystyle x}
is an adherent point of
S
{\displaystyle S}
in
X
{\displaystyle X}
if and only if this is true of
x
{\displaystyle x}
in every (or alternatively, in some) topological superspace of
X
.
{\displaystyle X.}
= Adherent points and sequences
=If
S
{\displaystyle S}
is a subset of a topological space then the limit of a convergent sequence in
S
{\displaystyle S}
does not necessarily belong to
S
,
{\displaystyle S,}
however it is always an adherent point of
S
.
{\displaystyle S.}
Let
(
x
n
)
n
∈
N
{\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}
be such a sequence and let
x
{\displaystyle x}
be its limit. Then by definition of limit, for all neighbourhoods
U
{\displaystyle U}
of
x
{\displaystyle x}
there exists
n
∈
N
{\displaystyle n\in \mathbb {N} }
such that
x
n
∈
U
{\displaystyle x_{n}\in U}
for all
n
≥
N
.
{\displaystyle n\geq N.}
In particular,
x
N
∈
U
{\displaystyle x_{N}\in U}
and also
x
N
∈
S
,
{\displaystyle x_{N}\in S,}
so
x
{\displaystyle x}
is an adherent point of
S
.
{\displaystyle S.}
In contrast to the previous example, the limit of a convergent sequence in
S
{\displaystyle S}
is not necessarily a limit point of
S
{\displaystyle S}
; for example consider
S
=
{
0
}
{\displaystyle S=\{0\}}
as a subset of
R
.
{\displaystyle \mathbb {R} .}
Then the only sequence in
S
{\displaystyle S}
is the constant sequence
0
,
0
,
…
{\displaystyle 0,0,\ldots }
whose limit is
0
,
{\displaystyle 0,}
but
0
{\displaystyle 0}
is not a limit point of
S
;
{\displaystyle S;}
it is only an adherent point of
S
.
{\displaystyle S.}
See also
Closed set – Complement of an open subset
Closure (topology) – All points and limit points in a subset of a topological space
Limit of a sequence – Value to which tends an infinite sequence
Limit point of a set – Cluster point in a topological spacePages displaying short descriptions of redirect targets
Subsequential limit – The limit of some subsequence
Notes
Citations
References
Adamson, Iain T., A General Topology Workbook, Birkhäuser Boston; 1st edition (November 29, 1995). ISBN 978-0-8176-3844-3.
Apostol, Tom M., Mathematical Analysis, Addison Wesley Longman; second edition (1974). ISBN 0-201-00288-4
Lipschutz, Seymour; Schaum's Outline of General Topology, McGraw-Hill; 1st edition (June 1, 1968). ISBN 0-07-037988-2.
L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, (1970) Holt, Rinehart and Winston, Inc..
This article incorporates material from Adherent point on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Kata Kunci Pencarian:
- Adherent point
- Accumulation point
- Isolated point
- Closure (topology)
- Adherence
- Infimum and supremum
- Derived set (mathematics)
- Regular conditional probability
- Bogomil (disambiguation)
- List of mathematical properties of points
