created *induced metric*, just for completeness

At *enriched category* it uses to say that

A Top-enriched category is a topological category.

But then at *topological category*, which redirects to *topological concrete category* it says that it

does

notmean Top-enriched category.

Of course for many people it does. But to get the $n$Lab entries straight, and to go along with the entry *simplicially enriched category*, I started an entry

just for completeness (and since I need the material elsewhere).

]]>I have edited at *Tychonoff theorem*:

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)

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 looked at *real number* and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:

]]>A

real numberis something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form anumber field, denoted $\mathbb{R}$. The underlying set is thecompletionof the ordered field $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ suprema for every bounded subset with respect to the natural ordering of rational numbers.The set of real numbers also carries naturally the structure of a topological space and as such $\mathbb{R}$ is called the

real linealso known asthe continuum. Equipped with both the topology and the field structure, $\mathbb{R}$ is a topological field and as such is the uniform completion of $\mathbb{Q}$ equipped with the absolute value metric.Together with its cartesian products – the Cartesian spaces $\mathbb{R}^n$ for natural numbers $n \in \mathbb{N}$ – the real line $\mathbb{R}$ is a standard formalization of the idea of

continuous space. The more general concept of (smooth)manifoldis modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (seespacetime), or at least in classical physics. See atgeometry of physicsfor more on this.

I added the definition and several references on higher dimensional knots under knot.

]]>I added to *cylinder object* a pointer to a reference that goes through the trouble of spelling out the precise proof that for $X$ a CW-complex, then the standard cyclinder $X \times I$ is again a cell complex (and the inclusion $X \sqcup X \to X\times I$ a relative cell complex).

What would be a text that features a *graphics* which illustrates the simple idea of the proof, visualizing the induction step where we have the cylinder over $X_n$, then the cells of $X_{n+1}$ glued in at top and bottom, then the further $(n+1)$-cells glued into all the resulting hollow cylinders? (I’d like to grab such graphics to put it in the entry, too lazy to do it myself. )

In the process of beginning to compile a list of central theorems in topology, on top of the list of basic facts in topology that I had been compiling the last days (of course there is some remaining ambiguity in which of these two lists to place a given item) I have created a stub for *Jordan curve theorem*.

I gave *locally compact topological space* an Idea-section and added the other equivalent definition (here).

at *vector bundle* I have spelled out the proof that for $X$ paracompact Hausdorff then the restrictions of vector bundles over $X \times [0,1]$ to $X \times \{0\}$ and $X \times \{1\}$ are isomorphic.

It’s just following Hatcher, but I wanted to give full detail to the argument of what is now this lemma.

]]>added a quick note on the CW-structure on real projective space: here.

]]>tried to polish one-point compactification. I think in the process I actually corrected it, too. Please somebody have a close look.

]]>brief note on *continuous field of C*-algebras*

added a little bit to *foliation*: a brief list of equivalent alternative definitions and and Idea-section with some general remarks.

the first paragraphs at *topological vector space* seem odd to me.

I’d think it should start out saying that a topological vector space is a vector space *over a topological field* $k$, such that etc.. Then the following remark presently in the entry, about the internalization using the discrete topology is moot.

I have spelled out the proof at *paracompact Hausdorff spaces equivalently admit subordinate partitions of unity*.

This uses Urysohn’s lemma and the shrinking lemma, whose proofs are not yet spelled out on the $n$Lab.

]]>For better readability, I have split off *proper map* (topology) from *proper morphism* (general) and added disambiguation. Added classes of examples at *proper map*.

I have created an entry on the *quaternionic Hopf fibration* and then I have tried to spell out the argument, suggested to me by Charles Rezk on MO, that in $G$-equivariant stable homotopy theory it represents a non-torsion element in

for $G$ a finite and non-cyclic subgroup of $SO(3)$, and $SO(3)$ acting on the quaternionic Hopf fibration via automorphisms of the quaternions.

I have tried to make a rigorous and self-contained argument here by appeal to Greenlees-May decomposition and to tom Dieck splitting. But check.

]]>I have added some minimum content to *Stiefel manifold*, also a little bit to *Grassmannian*

for completeness, I created *external tensor product of vector bundles*

I added the definition of uniform space in terms of covering families. But I don’t know the covering version of the constructive “axiom (0)”.

]]>wrote out statement and proofs at *shrinking lemma* (following Matt Rosenzweig here)

quick fix at suspension: distinction between plain and reduced/based suspension. More should be said here, but not by me right now.

]]>added to *compactification* the statement of the uniqueness of compactifications for “almost compact topological spaces”

Did anyone ever write out on the $n$Lab the proof that for $X$ locally compact and Hausdorff, then $Map(X,Y)$ with the compact-open topology is an exponential object? (Many entries mention this, but I don’t find any that gets into details.)

I have tried to at least add a pointer in the entry to places where the proof is given. There is prop. 1.3.1 in

- Marcelo Aguilar, Samuel Gitler, Carlos Prieto, sections 1.2, 1.3 of
*Algebraic topology from a homotopical viewpoint*, Springer (2002) (toc pdf)

but of course there are more canonical references. I also added pointer to

- Eva Lowen-Colebunders, Günther Richter,
*An Elementary Approach to Exponential Spaces*, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 (publisher)