How did the historical development of the notion of homeomorphism take place?
At first, I think the most important thing to study were continuous functions and metric structures.
When did the focus of mathematicians shift from metric to open sets?
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3This is already answered on Math Overflow, Homeomorphism historically. The source is Moore's paper The evolution of the concept of homeomorphism. – Conifold Jan 19 '21 at 11:45