# 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.

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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.

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?