# Closed subsets of a compact set are compact?

Suppose that K is a compact subset of a topological space X and A is a closed subset of X such that A is a subset of K. Prove that A is a compact subset of X.
I have pointed out the areas I dont understand - help would be great!
Take an open cover of A.
If you add the open set X\A to that, then you...
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Suppose that K is a compact subset of a topological space X and A is a closed subset of X such that A is a subset of K. Prove that A is a compact subset of X.

I have pointed out the areas I dont understand - help would be great!

Take an open cover of A.

If you add the open set X\A to that, then you have an open cover of K. - dont understand!

Since K is compact, there is a finite subcover. So that subcover also covers A.

Now you don't need the X\A set to cover E. - dont understand!

What's left is a finite subcover of the original cover that covers A

Thanks

I have pointed out the areas I dont understand - help would be great!

Take an open cover of A.

If you add the open set X\A to that, then you have an open cover of K. - dont understand!

Since K is compact, there is a finite subcover. So that subcover also covers A.

Now you don't need the X\A set to cover E. - dont understand!

What's left is a finite subcover of the original cover that covers A

Thanks

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