Is there a infinite set such that all* topologies on the set can be induced by a metric?

[gritty needle]

Prove or disprove the fact that there exists an infinite sized set such that any topological structure (except for the discrete topology) on that set can be induced by a metric.
(I.e. Using a metric d:X^2 -> R+ on the set X we can create the topology T = { D subset of X | For any a in D there exists an r > 0 such that Ball(a, r) subset of D } )
For example, the non-discrete topology on any set are induced by the discrete metric ( d:X^2 -> R+, d(x, y) = 0 if x = y and 1 if x != y )

[carmine adder]

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[gritty needle]

The picture above is only for clickbait purpose only. It has nothing to do with the question whatsoever

😉

[unreal gate]

cofinite topology on infinite set cannot be induced by a metric

@gritty needle

If we take two nonempty open sets U,V with empty intersection, then

$$ X = X\setminus (U\cap V) = (X\setminus U)\cup (X\setminus V) $$

[pure pagoda]

aL

[unreal gate]

but if X is infinite, it can't be a union of two finite sets

in other words, infinite cofinite space cannot be hausdorff

but metrizable topologies are most certainly hausdorff

[pure stag]

you mean R+U{0}