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4h
revised Definition of neighborhood and open set in topology
typo
1d
comment Fundamental groupoid
A more interesting topic is to use Moore paths in discussing paths in $X$: Such a path is a pair $(f,r)$ where $r \geqslant 0$ and $f:[0,\infty) \to X$ is a map which is constant on the interval $[0,\infty)$. So a Moore path $(f,r)$ is thought of as a journey of length $r$. Composition of such paths gives a category of paths in $X$. This type of structure, or a variation, is used in Crowell and Fox's book on Knot Theory, and in my book "Topology and Groupoids", and makes the theory clearer IMHO. The cogroupoid idea needs modification for this structure.
2d
revised Best book for topology?
changed a link, and added one
2d
revised Best book for topology?
changed a link
Apr
28
revised Products of quotient topology same as quotient of product topology
typos
Apr
25
revised Questions on CW complex structure
typo
Apr
25
answered Questions on CW complex structure
Apr
14
revised Dunce hat is simply connected
added a link to the source of the figure
Apr
12
comment Homomorphisms of Chain Complexes
This could be about the monoidal closed category of chain complexes, Old papers of mine (before that terminology was available) 3. Cohomology with chains as coefficients'', — Proc. London Math. Soc. (3) 14 (1964), 545-565. pdf file 4. On Künneth suspensions'', — Proc. Camb. Phil. Soc. 60 (1964) 713-720. pdf from groupoids.org.uk/publicfull.htm could be helpful. See also ncatlab.org/nlab/show/closed+monoidal+category
Apr
11
revised How does attaching a 1-cell to a path connected CW-complex affect the fundamental group?
added some extra remarks
Apr
8
awarded  Yearling
Apr
4
comment The functor $\pi_{1}: Esp_{*} \rightarrow Grp $ preserves coproduct?
If you drop the base point and consider the fundamental groupoid functor $\pi_1: Esp \to Grpd$ from spaces to groupoids then your formula is true and easy to prove. The more general setting allows one to consider the effect of identifications of distinct points, and deal with non pathconnected spaces.
Apr
1
comment How does attaching a 1-cell to a path connected CW-complex affect the fundamental group?
I am not sure how to interpret what seems to be a downgrading of a view that the unit interval naturally should have two base points. See also these comments of Grothendieck: mathoverflow.net/questions/220561/… .Answers of mine here and elsewhere are intended to help students to be aware of this alternative viewpoint in order to make their own judgement.
Mar
31
revised Intuition of the meaning of homology groups
corrected links
Mar
28
answered How does attaching a 1-cell to a path connected CW-complex affect the fundamental group?
Mar
27
comment What are the dangers of visual exposition of mathematics?
Actually the usual expositions of mathematics are visual, using words and symbols, written mainly on a line, in which the symbols have a well defined relation to those on the left and those on the right. Is this the only possible way? The brain certainly does not operate only in such a "serial" way. I was and am fascinated by the way higher groupoid theory requires 2-dimensional rewriting to prove theorems. How will we cope with a required 5-dimensional rewriting?
Mar
26
comment Is this a different proof of the fundamental group being abelian?
More background is in my paper with Chris Spencer , ``$\cal G$-groupoids, crossed modules and the fundamental groupoid of a topological group'', Proc Kon. Ned. Akad. v. Wet. 7 (1976) 296-302, available from my publication list.
Mar
25
comment Recovering CW complex structure from a space
A CW-structure determines a filtration of a space, i.e. an increasing sequence of subspaces, namely the skeleta of the structure. We have advertised in a joint book "Nonabelian Algebraic Topology" (EMS, 2011) an approach looking first at homotopical defined algebraic invariants of filtered spaces.
Mar
21
revised study topology: homotopy and homology
added a link and comment
Mar
21
revised What are the dangers of visual exposition of mathematics?
changed links