# Homework Help: Funtion continuity and open sets

1. Apr 11, 2010

### complexnumber

1. The problem statement, all variables and given/known data

Suppose that $$f : (X,d_X) \to (Y,d_Y)$$. If $$f$$ is continuous,
must it map open sets to open sets? If $$f$$ does map open sets to
open sets must $$f$$ be continuous?

2. Relevant equations

3. The attempt at a solution

The answer to the first question is yes. The answer to the second question I guess is "no". Is this correct? How can I prove it?

2. Apr 11, 2010

### HallsofIvy

Neither is true. For example, a constant function is trivially continuous but maps any set, including any open set, to a singleton set which is not, generally, open.

Conversely, if B has the discrete topology, so that all sets are open, and A is a [0, 1] with the usual topology, the function f(x)= a for all $0\le x< 1/2$ and f(x)= b for all $1/2\le 0\le 1$, where a and b are distinct points in B, is not continuous but maps open sets to open sets.

Last edited by a moderator: Apr 11, 2010