For a set X with the discrete topology, show that for every $A\subset X$:
- $\text{int} A = A$
- $\text{ext} A = X\setminus A$
- $\partial A = \emptyset$
where int means the interior of $A$, ext means the exterior of $A$, and $\partial A$ is the boundary of $A$.
Well, I try by the open sets but I got nowhere…
Then I try by neighborhood and I got nowhere…
Of course that the proof of the boundary of A is really obvious, but I can't do the int….can somebody help?
Best Answer
The interior of a set S is the set of all points of S that are in some open set contained in S. But every set is open (in the discrete topology), so if x is in S, then the set { x } is an open set contained in S.