Clopen Sets: Closure = Interior?

[blahblah8724]

For a subset which is both closed and open (clopen) does its closure equal its interior?

 

[quasar987]

1) Prove:

a) A set is open iff it is equal to its interior.

b) A set is closed iff it is equal to its closure.

2) Conclude.

 

[blahblah8724]

But one of the questions in my example sheet said think of an example of a disconnected subspace T of a topological space S for which there are no nonempty subsets A,B of T such that A\cup B = T but \bar{A} \cap \bar{B} =∅

Surely if \bar{A} = A = A^o then \bar{A} \cap \bar{B} = A \cap B = ∅ which is the definition of disconnected so there are no examples?

 

[Office_Shredder]

The key is that you're looking at a subspace, the closure of A may be larger than A in S

 

[blahblah8724]

Do you mean larger than T?

 

[Office_Shredder]

For example if S is the real numbers and A is (0,1), then the closure of A is [0,1], which of course is not equal to A

 

[blahblah8724]

But how could you possibly go about proving that there are NO nonempty subsets

 

[Office_Shredder]

It helps if you have the actual example that you're going to work with. As you've observed this phenomenon is atypical behavior of disconnected spaces, so you really need to exploit the fact that you're working in a larger topology.

Think a bit about the example I gave with A... can you think of a disconnected subset T in R such that when you take the closure of its two separated halves, they intersect (hint: this is the same as just taking the closure of T)

 

[blahblah8724]

How about the subset (0,2) where the two halves are (0,1) and (1,2)? So the closure would be [0,1] and [1,2] which intersect at 1?

 

[blahblah8724]

Surely the union of those two halves don't make the entire space as they miss out the point 1?