Compact, connected, closed sets

  • #1
1. If set A is compact, show that f(A) is compact. Is the converse true?

2. If set A is connected, show that f(A) is connected. Is the converse true?

3. If set B is closed, show that B inverse is closed.

Any help with any or all of these three would be greatly appreciated.

Stumped!
 

Answers and Replies

  • #2
cristo
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Welcome to the forums.

Note that, for homework questions, you need to show some work before we can help you. For example, a good way to begin would be to state the definitions of compact, closed,.. and all the other terms you are using here.
 
  • #3
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If you're trying to prove these for arbitrary functions, good luck. I imagine that the problem stated that f is continuous.
 
  • #4
morphism
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Also, by "B inverse" you probably meant "f-1(B)", i.e. the preimage (or inverse image) of B under f.

So for future reference, it would be good if, in addition to showing your work and thoughts, you posted the problems correctly!
 
  • #5
Sorry, here goes.
1. If set A is compact, show that f(A) is compact. Is the converse true?
Ans. f:M to N is continuous and A subset M is compact. Then f(A) is compact.
The converse is not necessarilly true. For Ex: F(x)=0 for every x in R(real #'s) and
k={0}. Then f^-1(k)=R is not compact.

2. If set A is connected, show that f(A) is connected. Is the converse true?
Ans. f:M to N is continuous and A subset M is connected. Then f(A) is connected.
The converse is not necessarilly true. For Ex: F(x)=x^2 and k=1.
Then f^-1(k)={-1,1} which is not connected.

3. If set B is closed, show that B inverse is closed.
Ans. f is continuous on B if f is continuous on every x sub 0 element B.
a. f is continuous on B
b. for every x sub n to x sub 0 in A. f(x sub n) approaches f(x sub 0)
c. for any u open in N, f^-1(u) is open in M
d. for any F closed in N, f^-1(F) is closed in M.


I hope this is better. If anyone can add to this I would be greatfull.
 
  • #6
Office_Shredder
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Your answers for (1) and (2) are essentially just rewording the question. I'm not sure what the answer to three is supposed to be.

To start 1, write down the definition of compact, then suppose f(A) is not compact and see what you can find
 
  • #7
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You just reposted the question? Seems somewhat unnecessary...
 

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