# What is a compact space?

According to definition, a compact set is one where every open cover has a finite sub-cover.
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?

George Jones
Staff Emeritus
Gold Member
But since C2 is an open cover then there is a finite subcover C3 which is subset of C2.

Yes, but C3 can be the same as C2, it doesn't have to have smaller cardinality.

I see. So you mean subset but not necessarily proper subset suffices?
I have gotten confused by the symbol.

$$\subset$$

Does it mean subset or proper subset?

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This was exactly the same difficulty that I had with the notion of a compact set. I originally though a compact set meant a single point! But it doesn't.

$$\subset$$

Does it mean subset or proper subset?
In modern notation, it would mean a proper subset, but much of the older texts and definitions of topology use it in its more ambiguous meaning as simply a "subset", proper or equal.

A more straightforward definition of compactness is simply to say that:
A compact set is one where every open cover is either finite or has a finite sub-cover.

A compact set is an extention of the idea of a closed bounded set in spaces where neither close nor bounded makes much sense. In euclidean space, or spaces isomorphic to some euclidean space, compactness is equivilant to being closed and bounded.

I think the old notion of allowing a subset to be less than or equal to is a notion that really should be retired. It's confusing, especially in cases like this. It's continued survival is probably for reasons of ostentation rather than clarity.

"A compact set is one where every open cover is either finite or has a finite sub-cover." - Good definition!

That is one cursed problem in math for the new-comer.
The other one is the

$$\theta$$
$$\phi$$

Different authors define it differently.
I mean maths is supposed to be clear, logical and universal, how can that happen?

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I mean maths is supposed to be clear, logical and universal, how can that happen?
Pretension.