Show disjoint sets are mutually separated?

[cookiesyum]

Homework Statement

Show that any two disjoint nonempty open sets are mutually separated.

Show that any two disjoint nonempty closed sets are mutually separated.

Homework Equations

The Attempt at a Solution

Let A and B be two disjoint nonempty open sets:

Assume A and B share a boundary point.

That boundary point exists in A intersect B. If the sets are disjoint, they can't intersect so this is a contradiction.

Similar idea for closed sets.

Is this correct?

 

[LCKurtz]

No. What if A = (0,1) and B = (1,2). They both have 1 as a boundary point but it isn't in either.