Two of my favorite results, Urysohn’s lemma and Urysohn’s metrization theorem, deal with normal spaces. They have such beautiful proofshttps://twitter.com/TopologyFact/status/1120345310994235394 …
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The metrization theorem states that if you have a space that is second countable and normal, then it is metrizable. The upshot is that if your space is "nice enough", then we can describe the notion of convergence in the space with a distance function.
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An example of a topology that is not "nice enough" would be functions f:[0,1]->R under the topology of pointwise convergence. We cannot assign a distance function that induces the topology of pointwise convergence.
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