cianfa72
- 2,784
- 293
- TL;DR Summary
- About the properties of the initial topology defined on a set though the preimage of a non-injective map
Consider a non-injective map ##\pi## from a set ##M## to a set ##N##. ##N## is equipped with a topological manifold structure (Hausdorff, second-countable, locally euclidean).
Take the initial topology on ##M## given from ##\pi## (i.e. a set in ##M## is open iff it is the preimage under ##\pi## of an open set in ##N##). Such a topology on ##M## is second-countable, however is it Hausdorff ?
I believe it is not since points in ##\pi^{-1} (\{p \}), p \in N## are always in the same open set in ##M## (let me say there is not enough "resolution" in the initial topology on ##M## to be able to separate its points into disjoint open sets).
What do you think about ? Thanks.
Take the initial topology on ##M## given from ##\pi## (i.e. a set in ##M## is open iff it is the preimage under ##\pi## of an open set in ##N##). Such a topology on ##M## is second-countable, however is it Hausdorff ?
I believe it is not since points in ##\pi^{-1} (\{p \}), p \in N## are always in the same open set in ##M## (let me say there is not enough "resolution" in the initial topology on ##M## to be able to separate its points into disjoint open sets).
What do you think about ? Thanks.