# open and closed sets

• Mar 9th 2009, 02:42 PM
grandunification
open and closed sets
I need a proof that the empty set and the real set are the "only" sets that are both open and closed. Pretty easy to prove that they are open and closed, but I need to show that they are the only such sets that satisfy both properties.

Thanks,

Justin
• Mar 9th 2009, 03:38 PM
Plato
Quote:

Originally Posted by grandunification
I need a proof that the empty set and the real set are the "only" sets that are both open and closed. Pretty easy to prove that they are open and closed, but I need to show that they are the only such sets that satisfy both properties.

This question is perfect example of the limits of forums such as this.
We do not know what tools & background you have to help you.
On one level it is almost trivial.
Suppose that $A$ is a proper subset of real numbers which is both open and closed.
Then $\Re = A \cup A^c$ but $A^c$ would also be open because $A$ is closed.
That is a contradiction to the fact that $\Re$ is a connected space being the union of two nonempty disjoint open sets.

But do you have that result to use?