Can someone explain, why ,intuitively, the axioms of topology regarding open sets are the way they are? (i.e. that the infinite union of open sets must be open, and that the finite intersection of open sets is again open)?
How do these axioms "respect" out notion of "nearness"?
The intuition behind this definition becomes apparent after exposure to basic results in analysis. You study calculus and analysis in $\mathbb R^n,$ then later general metric spaces, and eventually it becomes clear that continuity can be defined without reference to a specific metric. This is a really weird and cool realization at first. But without some good background in more specific topics, students seeing the modern definition of a topological space are likely to be dazed and confused, and possibly in search of a different major.