# Find all the limit points and interior points (basic topology)

• nalkapo
In summary, for sets A={(x,y): 0<=x<=1, 0<=y<=1} and B={1-1/n: n=1,2,3,...}, the limit points are 1 and the interior points are all (x,y) values.
nalkapo

## Homework Statement

Find all the limit points and interior points of following sets in R2
A={(x,y): 0<=x<=1, 0<=y<=1} *here I used "<=" symbol to name as "less then or equal".
B={1-1/n: n=1,2,3,...}

## The Attempt at a Solution

the limit point of B is 1 as n goes to infinity. because for the limit point, in the neighborhood there must be infinitely many points.
and I think the limit points of A must be all (x,y) values, because in the neighborhood ve can find infinitely many points in the neighborhood of all points.
my problem is with interior points.
How can I approach in order to find interior points? and what are the interior points of these sets?

An interior point of a set A is a point, p, such that some neighborhood of p is contained in A. Look at, for example, points like (0, 1/2), or (1, 3/4), or (2/3, 1), etc. which lie on the lines bounding the rectangle. Any neighborhood must have some points on every side of the point and so some on the outside of the rectangle. Those cannot be interior points.

For the second case, you might want to actually write down some of the points:
0, 1/2, 2/3, 3/4, etc. Suppose you drew an interval, say, (1/4, 3/8) around 1/2 or (2/3- 1/24, 2/3+ 1/24) about 2/3? (1/24 is half of 1/12, the distance from 2/3 to 3/4.)

HallsofIvy said:
An interior point of a set A is a point, p, such that some neighborhood of p is contained in A. Look at, for example, points like (0, 1/2), or (1, 3/4), or (2/3, 1), etc. which lie on the lines bounding the rectangle. Any neighborhood must have some points on every side of the point and so some on the outside of the rectangle. Those cannot be interior points.

For the second case, you might want to actually write down some of the points:
0, 1/2, 2/3, 3/4, etc. Suppose you drew an interval, say, (1/4, 3/8) around 1/2 or (2/3- 1/24, 2/3+ 1/24) about 2/3? (1/24 is half of 1/12, the distance from 2/3 to 3/4.)

So, you mean (for B={1-1/n}) that for some points p, I cannot find a neighborhood which is a subset of B. for example in [1/4, 3/8] we cannot find any neighborhood which belongs to B. so, in B there is no interior points. Is that true?

## What is a limit point?

A limit point is a point in a set such that every neighborhood of the point contains at least one other point in the set.

## How do you find limit points in a set?

To find limit points in a set, you can start by identifying all the points in the set. Then, for each point, you can check if every neighborhood of that point contains at least one other point in the set. If so, that point is a limit point.

## What is an interior point?

An interior point is a point in a set that has a neighborhood entirely contained within the set.

## How do you find interior points in a set?

To find interior points in a set, you can start by identifying all the points in the set. Then, for each point, you can check if there is a neighborhood of that point that is entirely contained within the set. If so, that point is an interior point.

## What is the difference between a limit point and an interior point?

The main difference between a limit point and an interior point is that a limit point must have at least one other point in the set in every neighborhood, while an interior point must have a neighborhood entirely contained within the set. In other words, a limit point is a point on the boundary of the set, while an interior point is a point within the set.

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