Topology

by iarnab_kundu, Dec 12, 2018, 8:44 PM

For us here we consider $X,Y$ and $Z$ to be topological spaces, unless otherwise mentioned. Maps are NOT always assumed to be continuous unless otherwise specified.

Definition- We say $f:Y\to X$ is universally closed, if for every (Hausdorff?) base change $f\times\text{id}:Y\times Z\to X\to Z$ is a closed map.
Definition- We say that $f:Y\to X$ is a proper map, if pre-image of every quasi-compact subset of $Y$ is quasi-compact.

Proposition- (If $Y$ is Hausdorff?) $f$ (continuous?) is universally closed if and only if it is proper.

Definition- We say $f:Y\to X$ is universally open, if for every (Hausdorff?) base change $f\times\text{id}:Y\times Z\to X\to Z$ is an open map.





Proposition-

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This blog reflects my thoughts on the mathematics that I grapple with. Hopefully these rumblings could be organized as to be palatable to a mathematical audience.

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