I Why Are Topology Axioms Defined the Way They Are?

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Is there a way we can see why the axioms defining a topology/ topological space are the way they are?
 
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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.
 
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Thread '2-sphere intrinsic definition by gluing disks' boundaries'

A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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