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.}
| Proof |
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By assumption, S ⊆ X ⊆ Y {\displaystyle S\subseteq X\subseteq Y} and x ∈ X . {\displaystyle x\in X.} Assuming that x ∈ Cl X S , {\displaystyle x\in \operatorname {Cl} _{X}S,} let V {\displaystyle V} be a neighborhood of x {\displaystyle x} in Y {\displaystyle Y} so that x ∈ Cl Y S {\displaystyle x\in \operatorname {Cl} _{Y}S} will follow once it is shown that V ∩ S ≠ ∅ . {\displaystyle V\cap S\neq \varnothing .} The set U := V ∩ X {\displaystyle U:=V\cap X} is a neighborhood of x {\displaystyle x} in X {\displaystyle X} (by definition of the subspace topology) so that x ∈ Cl X S {\displaystyle x\in \operatorname {Cl} _{X}S} implies that ∅ ≠ U ∩ S . {\displaystyle \varnothing \neq U\cap S.} Thus ∅ ≠ U ∩ S = ( V ∩ X ) ∩ S ⊆ V ∩ S , {\displaystyle \varnothing \neq U\cap S=(V\cap X)\cap S\subseteq V\cap S,} as desired. For the converse, assume that x ∈ Cl Y S {\displaystyle x\in \operatorname {Cl} _{Y}S} and let U {\displaystyle U} be a neighborhood of x {\displaystyle x} in X {\displaystyle X} so that x ∈ Cl X S {\displaystyle x\in \operatorname {Cl} _{X}S} will follow once it is shown that U ∩ S ≠ ∅ . {\displaystyle U\cap S\neq \varnothing .} By definition of the subspace topology, there exists a neighborhood V {\displaystyle V} of x {\displaystyle x} in Y {\displaystyle Y} such that U = V ∩ X . {\displaystyle U=V\cap X.} Now x ∈ Cl Y S {\displaystyle x\in \operatorname {Cl} _{Y}S} implies that ∅ ≠ V ∩ S . {\displaystyle \varnothing \neq V\cap S.} From S ⊆ X {\displaystyle S\subseteq X} it follows that S = X ∩ S {\displaystyle S=X\cap S} and so ∅ ≠ V ∩ S = V ∩ ( X ∩ S ) = ( V ∩ X ) ∩ S = U ∩ S , {\displaystyle \varnothing \neq V\cap S=V\cap (X\cap S)=(V\cap X)\cap S=U\cap S,} as desired. ◼ {\displaystyle \blacksquare } |
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
- 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.
References
Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. ↩