Compact Topological Spaces

So it's taken me a few days to put this together, in that time i think i've come to grasp the definition of a Compact Topological Space (as given by Guitarist). But i've been trying to figure out how we prove that a topological space is compact.

In my studies of various other sources it's been stated that if a topological space can be mapped onto a closed and bounded subset of then is compact. To support this statement i guess it's necessary to show that; Closed and bounded subsets of are compact and that the image of a compact space is also compact, provided that the mapping is a bijection. But i'm having trouble following some of the various proofs around on the internet.

That a closed subset of would be compact seems intuitive enough, but i want to understand how we show this in a mathematically rigorous way. The definition of a compact space might come in handy, i'm guessing, so we would be required to show that there is a finite cover of the closed subset in question and that this is a subcover of .