Continuous mapping - closed, open, compact, bounded
(I'm sorry to post in one thread, but I didn't want to cluster up the forums with multiple threads)
The following are 8 possible outcomes of continuous mappings and its image and preimages. I have several questions regarding notations as well.
Let and be continuous.
1. If A is closed, is F(A) closed?
False. Let and define . A is closed in R but is open and not closed.
2. If A is open, is F(A) open?
I believe this is true but I am not sure.
3. If A is compact, is F(A) compact?
True. Let be a sequence in F(A). Let . Since A is sequentially compact, then there is a subsequence . Thus,
4. If A is bounded, is F(A) bounded?
False. Let and . Since is bounded, but is unbounded.
**If suppose it was is bounded, then would be bounded?
5. If A is closed, is closed?
I believe this is true. Since . So, if A is closed, then is open. If f is continuous, then the preimage of every closed set in R is closed in A. By definition.
6. If A is open, is open?
True. If f is continuous, then the preimage of every open set in R is open in A by definition.
7. If A is compact, is compact?
False, let A = [0, 1] which is compact because it is closed and bounded. Then let f(x) = 1/2, the the image which is not compact.
8. If A is bounded, is bounded?
I do not know. I believe this is false, but I can not think of a counterexample.
Regarding notations, what happens if instead of is continuous, it was is continuous. Would the results change?
Thank you for reading. Any help is greatly appreciated.