(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.

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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, isclosed?

I believe this is true. Since . So, if A is closed, then is open. Iffis continuous, then the preimage of every closed set in R is closed in A. By definition.

6.If A is open, isopen?

True. Iffis continuous, then the preimage of every open set in R is open in A by definition.

7.If A is compact, iscompact?

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, isbounded?

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.