Finding the set of interior points, the closure, and an example

[splash_lover]

Suppose that S=[0,1)U(1,2)

a) What is the set of interior points of S?

I thought it was (0,2)


b) Given that U is the set of interior points of S, evaluate U closure.

I thought that U closure=[0,2]


c) Give an example of a set S of real numbers such that if U is the set of interior points of S, then U closure DOES NOT equal S closure

This one I was not sure about, but here is my example:
S=(0,3)U(5,6) S closure=[0,3]U[5,6]
U=(0,6) U closure=[0,6]

d) Give an example of a subset S of the interval [0,1] such that S closure=[0,1].

I said if the subset S=(0,1/2)U(1/2,1) then S closure=[0,1]

Are my answers right for these? If not could you please explain what the answer is in detail?

 

[radou]

If S=[0,1)U(1,2), as you wrote, is (1/2, 3/2) contained in S? (Note that 1 is an element of (1/2, 3/2)).

 

[splash_lover]

I think so, when I read the problem that's all it had was S=[0,1)U(1,2). So I am assuming (1/2,3/2) is contained in S.

 

[radou]

OK, assume (1/2, 3/2) is an open subset of S. Since 1 is an element of (1/2, 3/2), can this statement be true?

 

[splash_lover]

no it can't because 1 is not included in S

 

[radou]

OK, so, can (0, 2) be the interior of S?

 

[splash_lover]

no, (0,2) can't be the interior of S. So it would be (0,1)U(1,2)?

 

[radou]

Yes. The closure of S you wrote down is correct. Note that a useful fact about closures is that a point is in the closure of a set if and only if every neighbourhood of that point intersects the set of interest. So, 1 is in the closure of S.

 

[splash_lover]

So the closure is [0,2].

Was the example i gave for part C correct?

 

[splash_lover]

Is it even possible to find an example for part c? I know the example I gave is wrong.

 

[radou]

Can it be correct, considering all we said in between?

 

[splash_lover]

no, because some of the points in U (set of interior points)are not included in the original set S.