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For #1, an example is given by two Moore spaces $M({\mathbb Z}/p^2,k)$ and $M({\mathbb Z}/p^3,k)$; the only cohomology is in degree $k$ and $k+1$ in characteristic $p$, and the ring and Steenrod modul …

Here's one example. Take any ring $R$ equipped with the following two operations: $$(x,y,z)\mapsto x+y-z$$ $$(x,y,z,w)\mapsto x+(y-z)(w-x)$$ It is easy to see that if you add a constant symbol $0$, …

Any compact group with no nontrivial normal closed subgroups has this property, since there can be no coarser Hausdorff topology and in any coarser non-Hausdorff topology the closure of the identity w …

I'm not sure what you mean by "coherent sheaf", as that term is usually only used in the presence of something like a complex structure. But by this answer, the cohomology of any sheaf vanishes in de …

Here's another proof, which shows that any connected paracompact locally Euclidean space X is second-countable. Cover X by Euclidean charts and take a locally finite refinement. Say an open set is g …

It's a standard result that the continuous locus is always $G_\delta$. For each $r>0$, let $U(r)$ be the set of points $x$ such that some neighborhood of $x$ maps into some ball of radius $r$. Then …

This isn't an answer but an argument that there isn't really a good answer. Having done a good amount of set theory and seen how you prove some of these statements to be independent, I tend to be rat …