- #1
cianfa72
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- 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 from a set to a set . is equipped with a topological manifold structure (Hausdorff, second-countable, locally euclidean).
Take the initial topology on given from (i.e. a set in is open iff it is the preimage under of an open set in ). Such a topology on is second-countable, however is it Hausdorff ?
I believe it is not since points in are always in the same open set in (let me say there is not enough "resolution" in the initial topology on to be able to separate its points into disjoint open sets).
What do you think about ? Thanks.
Take the initial topology on
I believe it is not since points in
What do you think about ? Thanks.