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.

Images:

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?

Preimages:

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.