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As you may have seen from watching the logs, I am beginning to write a page Introduction to Topology. This is meant to be in the style as the previous Introduction to Stable homotopy theory, but now for basic point-set topology, starting from scratch, with some motivation taken from analysis, and ending with basic covering space theory.
I’ll be developing this during the next months. At the moment it is skeletal. Comments are most welcome.
Thanks to those who fixed formatting on the page, over the weekend.
Due to the way I am using this page at the moment, I would kindly like to ask everyone to please alert me of each edit you make to the page, even if tiny. Better yet, don’t touch this page for the time being, but alert me of everything that you would want to edit. Thanks.
I would like to fix braces around and inside of norms. (Most are there; only a few are missing.)
Thanks for the alert. I think I have fixed the remaining ones now.
There are still a few places where \Vert - \Vert
appears but should be \Vert {-} \Vert
. (Compare $\Vert - \Vert$ vs $\Vert {-} \Vert$.)
Normally, I wouldn't say anything, I'd just add the braces, but …
Then I must be missing something after all. I thought I need to make sure to type
{\Vert - \Vert}
.
You need the outside braces if there is nearly anything next to the norm (so I always put these in unless the norm is the only thing inside the dollar signs); but you also need inside braces if there is an operator (such as $-$) inside the norm (which there usually isn't, so I leave those out most of the time). So, $\Vert x \Vert$
, $2 {\Vert x \Vert}$
, $\Vert {-} \Vert$
, and $2 {\Vert {-} \Vert}$
all use minimal braces.
I see, thanks. Okay, I’ll fix this.
Typos:
streetching; continous; Parts of topology is; Among the separation axiom; the study if the; P(X) (in Example 1.8); halfopen (in Example 1.10)
Thanks! Fixed now.
Definition 1.23, $\tau_1, \tau_2 \subset P(X)$ rather than $\tau_1, \tau_2 \in P(X)$
corarser
Example 1.26, $\ast \coloneqq (\{x\}, \left\{ \emptyset, \{x\}\right\})$, $x$ should be $1$.
Example 1.27, \tau = \left{ \emptyset, {0}, {1} {0,1} \right}, missing comma.
Example 1.29, \tau \coloneq P(S), should be coloneqq.
Thanks once more! Fixed now.
I am closing in on declaring “part I” finished: Introduction to Topology – 1 Point-set topology
I am beginning to seriously work on “part II”: Introduction to Topology – 2 Basic homotopy theory.
As previously for the first part, I would like to ask that for the time being you do not touch this entry, not even for trivial edits, until I declare my edits to be essentially done.
If there is anything that stikes you as in need of edit, please drop me a message here instead. Just for the time being.
I’ll be developing this during the next months. At the moment it is skeletal. Comments are most welcome.
Just a general comment: like mentioned elsewhere, an awareness of, and freely drawing from, tom Dieck’s recent textbook “Algebraic Topology” seems to be in the spirit of the nLab. He emphasizes groupoids, for example, making this book a counterexample to the “rule” (that I have seen R. Brown deplore somewhere—I do not recall where) that even today most books in the field tend to ignore groupoids.
I think Ronnie’s observation would have been that ’many’ books in the field tend not to use groupoid methods. His work has shown how groupoid methods are often much easier and more elegant to use than single based pointed spaces and groups and that is consistent with the view throughout the nLab pages I think. The other methods are still important however. To go further in that direction here would be leading ’off thread’.
I am about to declare Introduction to Topology – 2 to be essentially done.
Presently it ends with the computation of $\pi_1(S^1)$. Originally I had the ambition to write out more examples. But I am exhausted and about to take a week off for a conference. So let that be it for now.
I imagine one of the harder things to do when writing a book is deciding once and for all it is done.
Congrats on your achievement, Urs.
Thanks Todd.
What was interesting was the contrast: in parallel to me writing out these notes reviewing how traditional mathematics climbs up from the foundations to the concept of tangent bundles, I advised Felix Wellen on the finalization of his thesis which works out how “differentially cohesive homotopy type theory” does the analogous ascent.
So the scope is pretty much the same, but the means are of most different flavor.
Also it made me wonder: Felix’s Wellen’s theory is truly constructive. But do traditional constructive/intuitionistic versions of topology ever arrive at standard constructions, such as, say, the frame bundle of a manifold? I ended up getting the impression that before traditional constructive maths gets there, it is being side-tracked by a bunch of issues. But I may be wrong.
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