The usual idea of compactness of a space is that if the space has an open cover then it has a finite subcover which is not very intuitive. In this talk we attempt to look at some equivalent characterisations of compact spaces. At first we prove some important connections between compactness , totally boundedness and sequential compactness using a series of theorems that begin by proving that every compact metric space has the Bolzano Weierstrass property, move on to the Lebesgue covering lemma and then end by proving that every sequentially compact metric space is totally bounded and is compact.
The second part of the talk attempts to apply these ideas to a proof of Ascolis theorem the statement of which is given : If $X$ is a a compact metric space then a closed subspace of $C(X,R) or C(X,C)$ is compact iff it is bounded and equicontinuous. Along the way. We develop certain important ideas such as a metric space is compact iff it is complete and totally bounded. We conclude by mentioning some important applications of Ascolis theorem, for example Prokhorov's theorem in probability.