Today we will discuss compactness in the metric setting. Why metric? Because metric spaces lend themselves more easily to visualisation than other spaces.

Let us imagine a metric space $X$ with points scattered all over it. If we can find an infinite number of such points and construct disjoint open sets centred on them, then $X$ cannot be compact.

Hence what does it mean to be compact in a metric setting?

Compactness implies that an infinite number of points can’t be ‘far’ away from each other. There can only be a finite number of “clumps” of points such that each neighbourhood, however small, contains an infinite number of such “clumped-together” points. So should you peer at one cump through a microscope, however, strongly you magnify the clump, you will not see discrete points. You will see an impossibly dense patch that shall remain a solid continuus clump of points.