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Link between a topological space and a manifold
This doesn't make sense! The manifold $M$ comes with a topology i.e. with the collection of subsets $T$.
Can we define a induced metric like this?
The induced metric is just the restriction of $g$ to the tangent space of the submanifold. Or the restriction of $h$. The $h$, if you raise one index, is the projection operator.