Example 1

Every finite space is quasi-compact, and more generally every space in which there is only a finite number of open sets is quasi-compact. A finite space is compact if and only if it is discrete, for a finite Hausdorff space is discrete ($§ 8$, no. 1, Corollary to Proposition 3). Conversely, every compact discrete space is finite, for in such a space the sets consisting of a single point are open; hence the space is finite by $(C''')$.