I am having trouble understanding the definition of a compact topological subset..

This is defined as a subset of a topological space where every open cover has a finite subcover...

however as a subcover is a subset of a cover, isnt every open cover a subcover of itself..

therefore isnt the definition eqivalent to saying that the set has a finite open cover, which is obviously not the correct definition....

can anyone see where i am going wrong??