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.

]]>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.

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)

Created a little entry *Vect(X)* (to go along with *Vect*) and used the occasion to give *distributive monoidal category* the Examples-section that it was missing and similarly touched the Examples-section at *rig category*.

I have split off *classical model structure on topological spaces* from the entry on “model structures on topological spaces”.

My aim is to have in this entry a detailed, self-contained and polished account of the definition of the *standard* or *classical* model structure, its verification and its key consequences.

I have added a fair bit of material today. Not done yet, but I have to call it quits now.

]]>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 have spelled out the proofs that over a paracompact Hausdorff space every vector sub-bundle is a direct summand, and that over a compact Hausdorff space every topological vector bundle is a direct summand of a trivial bundle, here

]]>I have been making trivial edits (adding references, basic statements, cross-links ) to *Hopf invariant* and a bunch of related entries, such as *Kervaire invariant*, *Hopf invariant one problem*, *Arf-Kervaire invariant problem*, *normed division algebra*.

created a minimum at *real homotopy theory*

I have briefly fixed the clause for topological spaces at *contractible space*, making manifest the distinction between contractible and weakly contractible.