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So let say I have C1, which is an open cover, I have C2 subset of C1 which is also an open cover. But C2 is finite.

But since C2 is an open cover then there is a finite subcover C3 which is subset of C2.

And so on and so forth , we will definitely end up with Cz which may only have one element. Then there will be no more subset of Cz, then how can there be any more subcover?

Isn't there a contradiction?