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deanslist1411
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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.
If anyone can add to this I would be greatfull.
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.