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 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 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.