Answer to Question #117896 in Differential Geometry | Topology for Sheela John

Question #117896
Prove or disprove every topological space is metrizible
1
Expert's answer
2020-05-25T20:29:34-0400

First we prove that every finite metric space is discrete. Indeed, let "(X,\\rho)" be a metric space, where "X" is finite.

Let "l(x)=\\min\\limits_{y\\in X\\setminus\\{x\\}}\\rho(x,y)" for every "x\\in X". Then "l(x)>0", and so "B_{l(x)}(x)=\\{x\\}" for every "x\\in X".

So we obtain that "(X,\\rho)" is discrete topological space.

But not every finite topological space is a discrete topological space, so not every topological space is metrizable.


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