# Boundaries and discrete spaces

1. Sep 15, 2010

### Riemannliness

Do sets in a discrete topological space have boundaries?

2. Sep 16, 2010

### Mandark

Every subset of a discrete topological space has empty boundary. To see this recall the definition of the boundary of a set A as the intersection of the closure of A, and the closure of the complement of A, in a discrete space all subsets are both open and closed, so the closure of A is simply A and the closure of A^c is A^c, their intersection is empty. This should also fit our intuition of what discrete spaces are, spaces where all points are isolated, such as Z as a subspace of R (in the standard topology).

3. Sep 16, 2010