ABSTRACT
Euclidean spaces property enables to demonstrate that certain objects exist without constructing them explicitly. it allows to establish existence of objects obtained via a limiting process that might be difficult or even impossible to be obtained otherwise. Motivated by the Euclidean Bolzano–Weierstrass property and its significance, mathematicians were interested in understanding and studying the class of topological spaces which possess this property, namely those privileged spaces in which any sequences has a convergent subsequence, thus coining the term sequentially compact topological spaces. Motivated by the Heine-Borel property which is purely topological and involves only open sets, the concept of the so-called compact topological spaces emerged, that is of those spaces for which every open covering has a finite sub-covering. Remarkably and contrary perhaps to first appearances, compactness is equivalent to sequential compactness in the context of metric spaces. This equivalence breaks down when going to more generalised space frameworks and understanding when these concepts coincide is an issue.
