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## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• I am splitting off Zariski topology from Zariski site, in order to have a page for just the concept in topological spaces.

So far I have spelled out the details of the old definition of the Zariski topology on $\mathbb{A}^n_k$ (here).

• Will put up some stuff about Dold’s trick of taking a not-necessarily point finite partition of unity and making a partition of unity. There is a case when I know it works and a case I’m really not sure about - I need to find where the argument falls down because I get too strong a result. I’ll discuss this in the thread soon, and port it over when it is stable.

• Wrote out a proof for paracompact Hausdorff spaces are normal.

(By the way, I also looked at TopoSpaces here to check what they offer, and am a bit dubious about their step 5. But maybe I am misreading it. In any case, I feel there is a simpler way to state the proof.)

• at separation axiom I have expanded the Idea section here, trying to make it more introductory and expository.

• I have edited support to say that in topology the support of a function is usually to be the topological closure of the naive support.

• I have added to Galois connection some more remarks to the Idea section, and expanded the Examples-section with the material that Todd wrote here.

• Late last night I was reading in Science of Logic vol 1, “The objective logic”.

I see that the idea of cohesion is pretty explicit there, not in the first section of the first book (Determinateness, which has the discussion of “being and becoming” that Lawvere is alluding to in the Como preface) but in the second section of the first book, “The magnitude”.

There the discussion is all about how the continuous is made up from discrete points with “repulsion” to prevent them from collapsing to a single and with “attraction” that keeps them together nevertheless.

This “attraction” is clearly just the same idea as “cohesion”. One can play this a bit further and match Hegel’s Raunen to formal expressions involving the flat modality and the shape modality pretty well. I made some quick notes in the above entry.

On the other hand, that section 1 about being and becoming seems to be more about the underlying type system itself. Notably about the empty type and the unit type, I think

• The usual notion of Peano curve involves continuous images of the unit interval, not the whole real line (which could be considered as well, of course).

So I made some adjustments and stated some relevant facts at Peano curve, with a few pointers to proofs and to literature.

• found it necessary to split off geometric realization of categories as a separate entry, recorded Quillen’s theorems A and B there

all very briefly. I notice that David Roberts has more on his personal web (have included it as a reference)

• I added a little bit to maximal ideal (first, a first-order definition good for commutative rings, and second a remark on the notion of scheme, adding to what Urs wrote about closed points).

The second bit is almost a question to myself: I don’t feel I really grok the notion of scheme (why it’s this and not something slightly different that’s the natural definition, the Tao if you like). In particular, it’s where fields – simple objects in the category of commutative rings – make their entrance in the notion of covering by affine opens that I don’t feel I really understand.

• I added a reference in the section on terminology to Makkai’s ’Towards a categorical foundation of mathematics’, where he defines what he calls the ’Principle of Isomorphism’. This is essentially what ’evil’ captures, I think, and it is handy to have a published version with a sensible name to which to refer people.

Here’s a wild thought: what about renaming the page principle of isomorphism and having evil redirect there. It would necessitate a rewrite of the page, but still contain material about the jokey names (evil, kosher etc). I recall that someone here told how some of these in-jokes are off-putting to outsiders or newcomers (Zoran, maybe?). Just an idea.

• I have added to alternative algebra the characterization in terms of skew-symmetry of the associator.

• I noticed that the entry analysis is in a sad state. I now gave it an Idea-section (here), which certainly still leaves room for expansion; and I tried to clean up the very little that is listed at References – General

• I have edited at Tychonoff theorem:

1. tidied up the Idea-section. (Previously there was a long paragraph on the spelling of the theorem before the content of the theorem was even mentioned)

2. moved the proofs into a subsection “Proofs”, and added a pointer to an elementary proof of the finitary version, here

Notice that there is an ancient query box in the entry, with discussion between Todd and Toby. It would be good to remove this box and turn whatever conclusion was reached into a proper part of the entry.

At then end of the entry there is a line:

More details to appear at Tychonoff theorem for locales

which however has not “appeared” yet.

But since the page is not called “Tychonoff theorem for topological spaces”, and since it already talks about locales a fair bit in the Idea section, I suggest to remove that line and to simply add all discussion of localic Tychonoff to this same entry.

• I have removed the following discussion box from stuff, structure, property – because the entry text above it no longer contained the word that the discussion is about :-)

[begin forwarded discussion]

+–{: .query} Mike: Maybe you all had this out somewhere that I haven’t read, but in the English I am accustomed to speak, “property” is not a mass noun. So you can “forget a property” or “forget properties” but you can’t “forget property.”

Toby: Well, ’property’ can be a mass noun in English, but not in this sense. Also, if we were to invent an entirely new word for the concept, it would surely be a mass noun. Together, these may explain why it's easy to slip into talking this way, but I agree that it's probably better to use the plural count noun here. =–

[end forwarded discussion]

• As an outcome of recent discussion at Math Overflow here, Mike Shulman suggested some nLab pages where comparisons of different definitions of compactness are rigorously established. I have created one such page: compactness and stable closure. (The importance and significance of the stable closure condition should be brought out better.)

• I did a little editing over at empty set; the query-box discussion of $0^0$ looked like it could be summarized with dispatch and relegated to a remark. Revert back or re-edit if you don’t like it.

• added in CW-complex in the Examples section something about noncompact smooth manifolds.

Eventually it would be good to state here precisely Milnor’s theorem etc. Googling around I seem to see a lot of misleading imprecision in the usual statements along these lines (on Wikipedia and MO) concerning the distinctions between countably generated and general CW-complexes and concerning homotopy equivalence vs weak homotopy equivalence.

• I have edited a bit at general topology, trying to stream-line for readability.

• Added to the entry fuzzy dark matter pointer to Lee 17 which appeared today on the preprint server. This is just a concise 2.5 page survey of all the available literature, but as such is very useful. For instance it points out this Nature-article:

• Hsi-Yu Schive, Tzihong Chiueh, Tom Broadhurst, Cosmic structure as the quantum interference of a coherent dark wave, Nature Physics 10, 496–499 (2014) (doi:10.1038/nphys2996)

which presents numerical simulation of the fuzzy dark matter model compared to experimental data.

• I began writing the article club, with more to come.