I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the additional structure of an open set of open subsets of these points), I'm really struggling to understand intuitively why this is needed, particularly in relation to manifolds used in physical applications such as GR?! I've heard explanations such as, a topology provides a primitive notion of geometry to a set without introducing additional structure such as a metric etc, and that it in some sense quantifies the "nearness" of points to one another (in the set) without having a notion of distance, in terms of neighbourhoods of points in the set, but these seem a little unsatisfactory to me. Does one require a topology to be defined on a manifold such that the is a well-defined why to "stitch together" local coordinate patches to construct a global structure that is the manifold? Is it also required to define a notion of continuity on the manifold?