Limit vs limit point. #limit #point #metric #space #convergent #cauchy
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In the previous section we see the convergent and Cauchy sequences in a metric space with their properties. We see that a limit point is a point to which a sequence is convergent. Also every convergent sequence in a metric space is a Cauchy sequence and the converse of the theorem i.e., every Cauchy sequence is not convergent.We illustrate it with an example by considering (0,1] as metric space and a sequence {1/n} which is Cauchy but not convergent in (0,1] in detail. In the following there is an introduction to few points and few definitions. Now take a look upon them.
Contents :
1. Limit vs limit point
2. Every where dense or dense set
3. Nowhere dense set
4. Boundary and boundary point of a set
1. Limit vs limit point:-
Some times we often use the terms limit and limit points. Are they both have same meaning? This is a confusion to every reader.
On the real line, for instance the constant sequence { -1, -1, -1,...} is converges to -1.
Here '-1' is called limit but not a limit point.
∴ '-1' is not a limit point.
NOTE :
1.A convergent sequence is not convergent " on its own "; it must converge to some point in the space.
2. A sequence may have a limit, but cannot have a limit point.
3. The set of distinct points of a sequence may have a limit point, but cannot have a limit.
4. If a convergent sequence in a metric space has infinitely many distinct points, then its limit is a limit point of the set of points of the sequence.
5. Let X be a complete metric space, and let Y be a subspace of X. Then Y is complete iff it is closed.
2. Every where dense:-
Let X be a metric space and A is a subset of X. A is said to be dense or every where dense if Ā=X.
For example:-
1. Consider the metric space X=R, the set of all real numbers
We know Q the set of all rational numbers is a dense subset of the real line R since closure(Q) =R.
2. Let X=[0, 1] and we have (0,1)⊂X.
Since closure (0,1) =[0, 1]=X, (0, 1) is dense in [0, 1].
* A subset of a metric space is dense iff it intersects every non-empty open set.
* For any subset A of a metric space is a dense subset of its closure.
3. Nowhere dense:-
A subset A of a metric space X is said to be nowhere dense if its closure has empty interior
i.e., A is nowhere dense if Int(Ā)=ⲫ.
∴ A is nowhere dense means it does not contain any open intervals.
Examples:-
1. Since Int(closure(Z))=ⲫ, Z is nowhere dense in R.
2. The cantor set is nowhere dense.
3. {1/n/nꜪ N} is nowhere dense in R.
4. {x/0<x<1} is nowhere dense in R.
5. Boundary and boundary point of a set:- Here we see the bounds of a set and the points which lies on the border of the set called them as the boundary and boundary points.
Boundary point:
Let X be a metric space and A be a subset of X. A point x in X is called a boundary point of A if each open sphere centred on x intersects both A and A'.
Boundary of a set:
The boundary of a subset A of a metric space X is the set of all its boundary points.
Basically the boundary of A is denoted by B(A).
Also the boundary of A is the intersection of closure (A) and closure (A').
* Any closed subset of a metric space X is the disjoint union of its interior and it's boundary i.e. if E is any subset in a metric space X , E= int(E)⋃ B(E).
* The boundary of a closed set is nowhere dense.
people also ask:
4. Is a closed sphere is a closed set?
5. What is meant by a dimension?
#limit #point #metric #space #convergent #cauchy
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