Metrizable implies normal
This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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For full proof, refer: Metrizable implies monotonically normal
The proof of this result actually shows something stronger: any metrizable space is monotonically normal. In other words, we can choose open subsets for the closed subsets in a monotone fashion.