I'm having a bit of difficulty seeing whether a function or its inverse preserves whether a set's image/pre-image remains open of closed. There are 5 parts for which I must either prove or give a counter-example. I cannot find an example for 3. and can't think of how to prove 4. I put all of them down though - just in case I messed anything up.
Let be continuous. Also, let and
1. If is open, then is open.
False: If for some constant , then its range is closed no matter the pre-image.
2. If is closed, then is closed.
False: Using and , then .
3. If is bounded, then is bounded.
False: But I'm having difficulty thinking of an example.
4. If is open, then is open.
True: But again, I'm not sure where to turn for the proof.
5. If is closed, then is closed.
True: If I had the above, I'd use its complement to show this one holds, but since I don't... Let and let be a sequence of elements from that converges to . Since is continuous, converges to . Since is closed, is in C. So, is in .