OPEN & CLOSED SETS #open #closed #sets
https://youtu.be/k6F-o4jy56o?si=8-m4XyaAyyV4MnWr
OPEN & CLOSED SETS
In this session we see open & closed sets with their properties.
OPEN SET:-
A non empty set A in the real line R is called an open set , every point in A is an interior point of A.
Examples:-
* (a, b) is an open set.
* [a, b] is not open since the end points a, b are not interior points.
* Every finite set is not open.
* The sets N, Z and Q are not open sets.
* The set R of all real numbers is open.
* The set of complex numbers |z|<1 is open.
* The set of complex numbers |z| ≤1 is not open.
* Every neighbourhood is an open set.
Properties of open sets:-
1. Every open set is a union of open intervals.
2. Union of two open sets is open.
3. Intersection of two open sets is open.
4. Union of any collection of open sets is open.
5. Intersection of a finite collection of open sets is open.
6. Intersection of an infinite collection of open sets is not open.
CLOSED SET:-
A set in R whose compliment is open is called a closed set.
In the other sense,
A set A in R called a closed set, A contains all of its limit points.
" Here what we see that a set is closed if it contains all of its limit points, not all the elements of A are limit points".
Examples:-
* Every closed interval [a, b] is closed.
* An open interval (a, b) is not closed.
* Q is not closed.
* A finite set is closed.
* N, Z & R are closed sets.
* The set of complex numbers |z|<1 is not closed.
* The set of complex numbers |z| ≤1 is closed.
* The empty set ⲫ is closed.
Properties of closed sets:-
1. Union of two closed sets is closed.
2. Intersection of two closed sets is closed.
3. Intersection of any family of closed sets is closed.
4. Union of a finite collection of closed sets is closed.
5. Union of any family of closed sets is not closed.
#open #closed #sets
#union #finite #intersection #interval
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