Taking introductory topology, I got the impression that the real line is the fundamental example of a topological space. I wouldn't be surprised if the open and closed intervals of $\mathbb{R}$ were the prototypical examples of open and closed sets, and I think many important topological properties---including compactness, connectedness, and Hausdorffness---first arose because you need them to prove obvious facts about $\mathbb{R}$ and its subsets.
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