Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have written an article closed-projection characterization of compactness, so as to record a nice way of proving the Tychonoff theorem that is due to Clementino and Tholen. It’s rather direct and elementary, which doesn’t involve ultrafilters or nets or any such machinery. This might make it a possibility for a strong undergraduate classroom. (Munkres also has a proof which I haven’t cross-checked; the one I wrote up involves a smidge of categorical terminology, notably inverse limits.)
I didn’t want to stick in the proof at Tychonoff theorem as that article might be getting a bit bloated, but I did link there to the new article.
I created Cantor space to record its definition as a locale, but goodness knows there is no end to what might be written about it.
I noticed that presently topological basis redirects to basis in functional analysis instead of to the entry topological base. This seems dangerous. I’d like to change that redirect.
At limits and colimits by example an entry for Kuratowski pair was requested, so I created it. The place where this should really be linked to is the entry ordered pair, so I added a link there. Also, I made that entry mention Cartesian product, which it didn’t.
I have given product topological space an semi-informal Idea-section that means to quickly and transparently tell the reader what they need to know.
This is in reaction to what we presently have at Tychonoff product which presently seems needlesly intransparent for the purpose of a reader who just wants to know what the open subsets actually are.
I was thinking about merging product topological space with Tychonoff product. I haven”t yet, but mostly just due to lack of energy.
The entry for infinitesimal extension said that an infinitesimal extension of rings was an epimorphism of rings with nilpotent kernel. I’ve changed this to say a quotient map of rings with nilpotent kernel. I hope this is correct: for example, localization maps are ring epimorphisms, and often have zero kernel (so in particular, nilpotent kernel) but geometrically these correspond to dense open inclusions, which are in no sense infinitesimal extensions.
Finally created stereographic projection. Lots more could be said, including metric/conformal aspects.
Somebody anonymous has created codiagonal comodule structure with no content. But let’s leave the it there and simply fill in some.
Presently I am concerned with the following: I want to teach some basics of limits and colimits of topological spaces to undergraduates. I had tried to gently introduce some category theoretic terminology as I introduced topological spaces as such, but a little testing reveals that part of the audience would rather not see these side remarks turn out to become required reading.
But now since all the shapes of diagrams that I’d be about to consider anyway are free, I figured I circumvent the need to speak of diagrams as functors by restricting to free domains and simply giving everything explicitly in components, with the underlying category theory again relegated to side-remarks that may be ignored at will.
I am trying my hands at an exposition of this kind in a new entry free diagram.
Eventually this kind of material might also be worthwhile as introductory exposition at limits and colimits by example.
I have a request to the logic experts:
The entry classical logic is a bit thin. I would like to be able to point to it so that readers who don’t already know about it all, get away with a decent idea of what is meant. Might somebody have the energy to add a few lines?
Of course I could add a few lines myself, but I imagine it would be more efficient if some expert here did it from scratch, instead of having to improve on what I would come up with.
Especially it would be nice if the following keywords were at least mentioned and maybe briefly commented on in the entry:
Also for instance the keyword constructive anything is presently missing from the entry.
Thanks!
had occasion to give attaching space its own little entry. Cross-linked with Top and CW-complex.
I added a sentence on ’eigenfunction’ to eigenvector.
Wrote about Conduche functors.
I have created an entry
to be used for inclusion in other entries, and I have included it into relevant entries (mostly in their subsections “Related entries”, except at Top and at limits and colimits by example).
it seems we were lacking order topology, so I created a minimum and cross-linked a bit.
partition of unity, locally finite cover
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.
I have touched the entry K-topology. Polished the definition and spelled out in some detail why it is not regular (while clearly Hausdorff).
The example which used to be in the entry (rational numbers with their subspace K-topology here) ends with
This space can be used to construct a quasitopological groupoid which isn’t a topological groupoid.
This statement should be accompanied with some reference. I suppose it refers to a construction that David R. (who wrote this back in 2010, rev #1) used in one of his articles?
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.)
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.
started a stub for topological invariance of dimension
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)
at schemes are sober I have added pointer to a comprehensive proof, here
gave finite intersection property an entry, added statement and proof of the corresponding characterization of compact topological spaces. Pointed to this from Tychonoff theorem, where the statement is used.
I added recollement as an example to the pages on reflective and coreflective subcategories.
I did a little editing over at empty set; the query-box discussion of 00 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 to Hausdorff topological space a brief paragraph Beyond topological spaces
I have substantially expanded the exposition at diagram. In the section Diagrams shaped like categories.
First I expanded on the quick functorial definition of diagrams and their ((co-)limiting) (co-cones) and then I added a detailed component-based definition meant for readers not fluent in unwinding the former definition.
Haven’t proof-read yet, as my battery is dying now.
I noticed that the entry disjoint union had no pointer to or from either of dependent sum, dependent sum type.
I have added the bare pointers now (no energy for more now). But this means some basic examples are missing in these entries.
at triangle inequality the discussion of the interpretation in enriched category theory had been missing. I have added in a corrresponding section here and cross-linked with Lawvere metric spaces.
added to interior the relation between topological interior and topological closure, here (also at topological closure, here)
I was un-graying some links at epsilontic analysis. Among the titles of non-existing entries that are still being requested is
“classical analysis”
“topological property”.
Is it likely/desireable that we will have entries with these titles? Maybe we should change these links to point to “analysis” and to “topology” instead?
started sidebar topology - contents, but am running out of time for this…
For completeness I created Kolmogorov space.
added statement and proof at compact Hausdorff spaces are normal.
Also added a pointer to this entry from the entries that previously claimed this without reference.
I started an entry for Rijke’s join of maps. How do you place the * lower?
I have added both to proof and to experiment pointer to
with the quote (from p. 2):
we claim that the role of rigorous proof in mathematics is functionally analogous to the role of experiment in the natural sciences
added statement and proof that sequentially compact metric spaces are equivalently compact metric spaces
added statement and proof at continuous images of compact spaces are compact. Cross-linked with extreme value theorem and other related entries.
added a proof at compact subspaces of Hausdorff spaces are closed
added statement-entry closed subspaces of compact Hausdorff spaces are equivalently compact subspaces.
with proof by pointing to the two directions:
Cross-linked with compact space, closed subspace and also with Heine-Borel theorem
created the line with two origins
I added a remark about the contravariant Yoneda embedding c↦C(c,−) to the page on the Yoneda embedding.
It’s pretty elementary, but I think worth mentioning for those new to category theory that this is just the Yoneda embedding of the opposite category Y:Cop→[(Cop)op,Sets]=[C,Sets].
added statement and proof that sequentially compact metric spaces are totally bounded
added statement and proof to Lebesgue number lemma.
did the same for sequentially compact metric spaces are equivalently compact metric spaces but now the nLab seems to have gone down before I could submit the edit.
I move null set to null subset and added more about how these are defined in unusual contexts.
I added the theorem that complete norms on a real vector space are unique (up to topological equivalence) at norm#dreamUnique. (This is false in classical mathematics, of course, but it’s true in dream mathematics.) Also true for F-norms.
expanded endomorphism operad
(it’s still a bit rough, but I am a bit rushed and have no time to polish)
I added to uniformly regular space a definition of “uniform apartness space” and a proof that under uniform regularity, these coincide with ordinary uniform spaces. I think this is interesting because it seems to be one of the purposes of uniform regularity (and local decomposability).
I rescued an empty page and wrote F-norm.
This theorem, with a constructive proof, is now at convergence space. (The usual proof in undergraduate metric-space theory uses both excluded middle and countable choice1, so I wrote this mostly to verify that it is actually perfectly constructive in the general setting.)
ETA: And the straightforward generalization to nonsequential spaces would use choice of arbitrarily high cardinality. ↩
Back in 2015, Bas Spitters wrote filter space. These are even more general than convergence spaces! (In a filter space, even if two filters both converge to the same point, their intersection might not.) I've put in the definition from the cited paper by Martin Hyland.
I added a note about this to Grothendieck topos. Also I fixed a mistake that Mike found in the classification at pretopos.
Since I wondered what they were, I started an entry Gorenstein ring spectrum, which then needed Gorenstein ring. Not sure I’m much the wiser as to their importance. There should be a lot to say about related duality.
created exact (infinity,1)-functor
created category of operators
I realized that we had a stub entry “configuration space” with the physics concept, and a stub entry “Fadell’s configuration space” with the maths concept, and no interrelation between them, also without any examples. So I created a disambiguation page
and then
but I also left
separate for the moment, thinking that in principle the term in matematics may be understood more generally, too. But maybe something should be merged here.
I added the example of the unordered configuration space of ℝ∞ as a model for the classifying space for the symmetric group to the relevant entries. But otherwise they do remain stubby, alas.
created law of double negation with just the absolute minimum. Added a link from double negation, but nothing more.
In the nLab article on the universal enveloping algebra, the section describing the Hopf algebra structure originally stated that “the coproduct Δ:UL→U(L∐L)≅UL⊗UL is induced by the diagonal map L→L∐L.”
I assume that this is a mistake, and I have since changed the coproduct ∐ to a product ×. However, I don’t know a great deal about Hopf algebras, so please correct me if I’ve made a mistake here.
In line with the “pages named after theorems” philosophy, I’ve created toposes are extensive, including in particular the (somewhat hard to track down) constructive proof that a cocomplete elementary topos is infinitary extensive.
Urs started pointless topology, and I continued it.