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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 25th 2010

created basis for a topology and linked to it with comments from coverage and, of course, Grothendieck topology

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 25th 2010

added the link to basis for a topology in the corresponding stub-subsection of the entry base.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeMay 25th 2010

There is also a calssical notion of a base of topology where topology is in the sense of classical topological structure.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeMay 25th 2010

There is also a classical notion of a base of topology where topology is in the sense of classical topological structure.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 25th 2010
• (edited May 25th 2010)

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.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeMay 25th 2010

Thanks. Should it be obvious to me that good open covers are even a coverage?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 25th 2010

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.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeMay 25th 2010

Are you assuming some sort of local contractibility of the spaces involved?

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMay 25th 2010

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.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 25th 2010

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.

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeMay 25th 2010

Okay, I believe it on Diff; I was confused because good open cover referred to arbitrary topological spaces.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMay 25th 2010

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.

• CommentRowNumber13.
• CommentAuthorDavidRoberts
• CommentTimeMay 26th 2010
• (edited May 26th 2010)

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)

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeMay 26th 2010
• (edited May 26th 2010)

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…

• CommentRowNumber15.
• CommentAuthorAndrew Stacey
• CommentTimeMay 26th 2010

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.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeMay 26th 2010

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?

• CommentRowNumber17.
• CommentAuthorMike Shulman
• CommentTimeMay 26th 2010

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.”

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMay 26th 2010

That sounds good. I have implemented that now.

David, is that allright with you?

• CommentRowNumber19.
• CommentAuthorDavidRoberts
• CommentTimeMay 27th 2010

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.

• CommentRowNumber20.
• CommentAuthorDavidRoberts
• CommentTimeMay 27th 2010

Just put a bit more at locally contractible space, including an example from MO I just asked for.

• CommentRowNumber21.
• CommentAuthorzskoda
• CommentTimeMay 27th 2010

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.