Question

By using the definition of a compact set in R, show that a closed subset X of a compact set K is compact.

Answer 1

haha good luck! :P

Answer 2

what is your definition of compact? every open cover can be reduced to finite sub cover?

Answer 3

if so, proof is not hard, and doesn't even depend on being in R
your set \(A\) s closed inside of a compact set C. if your open cover of is say \(\cup O_\beta\) then \(\cup O_\beta \cup A-C\) is an open cover of C and therefore can be reduced to a finite sub-cover.

Answer 4

Definition : A subset A of R is compact if every opening covering of A contains a finite subcovering of A

Answer 5

ok then proof above should work right?

Answer 6

noting of course that \(A-C\) is open since C is closed.