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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.

If anyone can add to this I would be greatfull.