Why Are Topology Axioms Defined the Way They Are?

[kent davidge]

Is there a way we can see why the axioms defining a topology/ topological space are the way they are?

 

[fresh_42]

One way is to consider it as a generalization of the metric induced topology in $\mathbb{R}^n$. It is what is left if we take away all the metric stuff and concentrate on what is really needed: open sets. Open sets are needed to define continuous functions, which would be an approach from the morphism point of view.