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created basis for a topology and linked to it with comments from coverage and, of course, Grothendieck topology
added the link to basis for a topology in the corresponding stub-subsection of the entry base.
There is also a calssical notion of a base of topology where topology is in the sense of classical topological structure.
There is also a classical notion of a base of topology where topology is in the sense of classical topological structure.
Okay, I merged what used to be “basis for a topology” into Grothendieck pretopology, putting suitable redirects and all (even successfully clearing the cache, now how is that?).
I added a mention of the counterexample of the good open cover coverage, which is not a pretopology.
Thanks. Should it be obvious to me that good open covers are even a coverage?
Should it be obvious to me that good open covers are even a coverage?
Every open cover may be refined by a good open cover, by choosing a good open cover of each of the open subsets that are part of the open cover.
Are you assuming some sort of local contractibility of the spaces involved?
Are you assuming some sort of local contractibility of the spaces involved?
Ah, yes. I mean, I am thinking of this as a coverage on CartSp, even.
Well, let’s see: at coverage in the Examples-section currently good open covers is listed as a coverage on Diff. I think that’s correct, but let me know if I am overlooking something.
Okay, I believe it on Diff; I was confused because good open cover referred to arbitrary topological spaces.
Right. I hope it doesn’t claim that on Top good open covers form a coverage, just that there is a notion of good open cover for any topological space.
I am in the process of checking some details. I’ll try to write a more comprehensive account on where good open covers form a coverage and where not a little later.
Can we get away with knowing that the inclusion map of each open is null-homotopic, instead of the open being contractible? This property is enough for applications like classifying locally homotopy trivial fibrations (cf Wirth-Stasheff’s JHRS paper)
null-homotopic, instead of the open being contractible?
So you define contractible here not as null-homotopic?
What can we say about this here: for $X$ a locally contractible topological space, let $OpC(X) \subset Op(X)$ be the full subcategory of open subsets on those that are contractible. Both are equipped with the coverage of good open covers.
Now, let’s see, $Sh(OpC(X))$ is still equivalent to $Sh(Op(X))$, I suppose, and…
There’s a difference between a space being contractible and a map being null-homotopic. The inclusion of a contractible subset is null-homotopic, but there are null-homotopic inclusions without the subspace being contractible, for example if the ambient space is contractible - so the inclusion of a circle in the plane is null-homotopic.
Ah, I see, you are thinking of null homotopy in the ambient space.
Hm, do we really not have terms to distinguish between these two notions at locally contractible space?
By analogy with “semi-locally simply connected” it seems that a natural phrase for admitting local opens whose inclusions are nullhomotopic would be “semi-locally contractible.”
That sounds good. I have implemented that now.
David, is that allright with you?
Yeah it’s good. I had something there, but it was a bit sketchy. I tidied up a bit where my old terminology conflicted with semi-locally contractible.
Just put a bit more at locally contractible space, including an example from MO I just asked for.
Well, Andrew, I dislike that you posted a link at ndigest to this page, as what I put in zoran: new entries are side topics which are not worthy to open a separate page, so it gives exactly opposite impression from what I am really focused on in nlab.
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