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## Main Question or Discussion Point

I've never actually seen a proof that a space is compact just from the definition. In metric spaces it was usually after the notion of closed and bounded or sequential compactness was introduced.

For example is there a way to prove [a,b] is compact (with the usual topology on the real numbers) just from definition? It seems almost impossible... you have to consider an arbitrary open cover!

Is it possible to use the fact that this topology has a natural correspondence with the metric space? Because open sets in the metric space are the same as the open sets in the topology?

For example is there a way to prove [a,b] is compact (with the usual topology on the real numbers) just from definition? It seems almost impossible... you have to consider an arbitrary open cover!

Is it possible to use the fact that this topology has a natural correspondence with the metric space? Because open sets in the metric space are the same as the open sets in the topology?