# Interior of a set

## Homework Statement

Consider the excluded point topology on a set X.
Determine Int(A) and Cl(A) for sets A containing p and for sets A not containing p.
Excluded point topology is all the subsets of X that exclude p.
where p is in X.

## The Attempt at a Solution

So the interior of a set A is the union of all open sets contained in A.
Would the interior for A be A-p , where we exclude p.
and would the interior be A where we include p.
Im not sure what the smallest closed set would be that contains A.
It seem like it would just be A.

## Answers and Replies

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SammyS
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Homework Helper
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## Homework Statement

Consider the excluded point topology on a set X.
Determine Int(A) and Cl(A) for sets A containing p and for sets A not containing p.
Excluded point topology is all the subsets of X that exclude p.
where p is in X.

## The Attempt at a Solution

So the interior of a set A is the union of all open sets contained in A.
Would the interior for A be A-p , where we exclude p.
and would the interior be A where we include p.
I'm not sure what the smallest closed set would be that contains A.
It seem like it would just be A.
It would be helpful to organize you answers better.

If p∈A,
Determine Int(A).

Determine Cl(A).​

If p∉A,
Determine Int(A).

Determine Cl(A).​

for the closure of those sets, should I try a proof by contradiction.
for the second one assume that p is not in the closure.Since p is not in A it is in the complement so it is in a closed set.

Have you proven any properties for closure and interior? For example, you can show that interior of A is the largest open set contained in A and closure of A is the smallest closed set which contains A. If you can prove this, then you're almost done.

SammyS
Staff Emeritus
Homework Helper
Gold Member
for the closure of those sets, should I try a proof by contradiction.
for the second one assume that p is not in the closure.Since p is not in A it is in the complement so it is in a closed set.
If $p\in\text{A}\,,$ then is set A open or is A closed?
If $p\notin\text{A}\,,$ then is set A open or is A closed?