INTERIOR AND LIMIT POINTS OF A SET : #interior #limit #point #of #set
The video of today's concept is available in the following link
https://youtu.be/HNMW09siyt4?si=-8i4Q0ofYkeM1Wbv
In mathetics, the terms the neighborhood of a point, interior and limit points have basic role to define open and closed sets.
Before going to open and closed sets , here we see the concepts of neighborhood , interior and limit point.
** N for set of natural numbers
** Z is for set of integers
** Q is for set of rational numbers
** R is for set of real numbers
NEIGHBOURHOOD OF A POINT
Let x be a point of a nonempty subset A of the real line R.
The neighborhood of x with radius ϵ is denoted by Nϵ(x) and is given by Nϵ(x) = {y Ꜫ A/|x-y|<r}
The neighborhood of x is also simply written as
nbd of x or nbd(x).
Also the neighborhood of x, Nϵ(x) is also called as ϵ-nbd of x.
Note :
1. An open interval (a,b) is a nbd of its points.
2. A closed interval [a,b] is a nbd of its points except for end points a and b.
3. The sets N,Z,Q are not nbd's of its points.
4. The set R of real numbers is a nbd of its points.
Interior point :
A point x of a subset A of real line R is called an interior point of A if there exists atleast one nbd, Nϵ(x) of x such that Nϵ(x) ⊂ A.
Note :
1. Every point of an open interval is interior point of the interval.
2. N,Z,Q have no interior points
3. Every point of R is an interior point.
Limit point of a subset of R:-
Definition 1:-
A point p Ꜫ R is said to be a limit point of a subset A of R, if every neighbourhood of p has a point of A other than p itself.
Definition 2.
A point p in R is said to be a limit point of a subset A of R if every nbd of p has infinite points of A.
Note :
1. The limit point a nonempty set A is need not belongs to A.
2.
If a point p is not a limit point of A iff there
exists a nbd of p which does not contain any point of A.
1. The null set ⲫ has no limit point
2. A finite set S has no limit point
3. The sets N & Z have no limit points
4. Every real number is a limit point of Q
5. Every real number is a limit point of R
6. Every point of S=(a,b) is a limit point of S
people also ask
what is an extended real number?
#interior #limit #point #set
#(a,b)
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