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Is there some general stuff about weak topologies - to support what is an ad hoc definition here? More often 'weak' is something like 'as subspace of a product' - is the point here 'as quotient space of a coproduct.
The article needs other work, and checking. There must be plenty now known about the purely categorical side of the CW complex homotopy category.
Charles
Ah, but Frank Adams used to be somewhat scathing about the simplicial techniques. For doing homotopy theory, at least. Charles Matthews 19:12, 4 Dec 2004 (UTC)
I propose that we make a separate page for the 'smash product' and kill the redirect to this page for that term. Smash products need not be defined only for CW-complexes, as far as I am aware, and there are some important examples for smash products that could be provided in a separate article. - Gauge 03:45, 18 Oct 2004 (UTC)
OK. While we're at it, pointed space needs a page. Hard to make exciting, perhaps - it's an example of a coslice category ?! Charles Matthews 11:42, 18 Oct 2004 (UTC)
In his algebraic topology book, Allen Hatcher takes a cell complex to be synonymous with a CW complex. Should we mention this in the article, or do most authors mean something more general by a cell complex (as the article currently indicates)? -- Fropuff 17:22, 2004 Dec 1 (UTC)
This, lightly amended, is one definition from a Google search:
A cell complex or simply complex in Euclidean space is a set of convex polyhedra (called cells) satisfying two conditions: (1) Every face of a cell is a cell (i.e. in ), and (2) Give two cells, their intersection is a common face of both. A simplicial complex is a cell complex whose cells are all simplices.
I don't doubt that the Hatcher definition is probably common usage in algebraic topology; but it doesn't seem safe to make it the WP definition. Charles Matthews 19:26, 1 Dec 2004 (UTC)
I'd like to suggest this article include a brief sketch of (at least the statements of) Whitehead's theorems about when you can replace a CW-complex by a simpler CW-complex, provided you know enough about the homotopy groups. Ie: n-connected implies there is a homotopy-equivalent CW-complex with a trivial n-skeleton, etc. I'd be happy to put them in, eventually. I primarily want these results to appear as motivation for the h-cobordism theorem, to be used in h-cobordism and the handle decomposition articles. Rybu ( talk) 23:30, 18 November 2009 (UTC)
This article really needs pictures. Lots of pictures. From cell attachments to example CW-complexes, CW-decompositions of surfaces, perhaps lens spaces, etc.
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Rybu ( talk) 01:11, 19 November 2009 (UTC)
Let me plus this more than 12 years old request. Arnaud Chéritat ( talk) 08:13, 24 June 2022 (UTC)
cell complex redirects here. Is it the same as a CW complex or is it ``too general``? -- MarSch 16:08, 7 November 2005 (UTC)
seems a bit doubtful to me, since the article does not (yet?) mention CW pairs ;) Still, i also created the redirect from CW pair as long as there is no article on the subject. - Saibod 09:31, 24 April 2007 (UTC)
In the section entitled "the homotopy category" appears the phrase "The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category". The identity of these experts should really be revealed. See Wikipedia:Avoid_weasel_words —Preceding unsigned comment added by 220.239.200.146 ( talk) 07:05, 24 August 2009 (UTC)
It's the second sentence of the article that worries me. It implies that CW complexes, presumably at the level of maps, have good categorical properties. It's rather far-fetched to say that. If you think of any desirable property for the category of CW complexes and maps to have, the chances are it hasn't got it. OK, it has finite sums and finite products. But limits, even finite ones? Colimits, then? Function objects? Mapping cones of all maps? Even claiming that CW's are better than simplicial complexes, categorically speaking, is doubtful. If it were categorical niceness that we wanted, we'd probably settle for simplicial sets — but they are less practical for the topologist, even if the motivic homologists love them.
We ought to substitute a better justification here: the sheer practicality of CW's for real-life calculations and manipulations. Ambrose H. Field ( talk) 16:40, 19 November 2009 (UTC)
I wonder if the homology/cohomology computations would be better kept in the cellular homology article? If we want to keep an example in this article perhaps we should make it a non-trivial one where the cellular attachment maps play a role in the computation. Or perhaps the cellular homology article should contain those types of examples, not this article. Rybu ( talk) 20:57, 22 November 2009 (UTC)
The further examples section has some problems. It says things like algebraic varieties are CW-complexes. This is false. A CW-complex requires a filtration by skeleta, attaching maps, etc. Algebraic varieties do not have those. I think what the author means to say is that they have the homotopy-type of cw-complexes. The same for the topological manifolds example. Rybu ( talk) 21:10, 22 November 2009 (UTC)
"The standard CW-structure on the real numbers has 0-skeleton the integers Z and 1-cells the intervals (...)"
Should it say, "has as 0-skeleton the integers and as 1-cells the intervals," perhaps? LokiClock ( talk) 09:08, 13 January 2010 (UTC)
Is the definition given in "Formulation" really complete? If yes, I would be interested to know what keeps me from constructing the following "CW complex", which, I think, should not be one:
Take the unit disk. Partition it as follows: The center plus all points on the boundary are the 0-cells. All the radii (without endpoints) are the (open) 1-cells. This fulfills all points of the definition, no? -- 84.75.56.196 ( talk) 16:36, 6 April 2012 (UTC)
I found a number of instances of the first notation below, and corrected it to read like the second one:
Notice that:
This stuff is codified at WP:MOS. Michael Hardy ( talk) 21:10, 16 May 2012 (UTC)
The reference to closure-finite redirects back to this article, perhaps a direct explanation of what "closure-finite" means should be included instead of linking a reference. — Preceding unsigned comment added by 130.237.198.164 ( talk) 10:13, 9 January 2013 (UTC)
It is written now: "An n-dimensional open cell is a topological space that is homeomorphic to the open ball." Therefore, 0-dimensional open cell is empty. (Because, the internity of a 0-dimensional ball is empty.) 91.77.190.196 ( talk) 09:27, 19 August 2014 (UTC)
What is the zero-dimensional open ball? What is the zero-dimensional (closed) ball? Why? 91.77.185.69 ( talk) 10:08, 19 August 2014 (UTC)
Hi! The zero-dimensional open ball and the zero-dimensional closed ball are both the one-point space, with the only topology that space can have. This is consistent with several definitions of "open ball" and "closed ball." For example, the open ball consists of all points in R^0 which are distance less than 1 from the origin; there is only one such point. As another example, the closed 0-ball is the one-point space, and its interior is therefore also the one-point space. (You should check this carefully from the definition of "interior.")
I posted this question on math.stackexchange, but it hasn't gotten much traction and perhaps it makes more sense to ask it here. Could anyone please give me some advice as to how to interpret the statement in the article that "trivalent graphs can be considered as generic 1-dimensional CW complexes"? What exactly is meant by the term "generic"? For example, are trivalent graphs characterized by some sort of universal property among 1-dimensional CW complexes? (Is this somehow implied by the statement about attaching maps?) Thank you, Noamz ( talk) 20:57, 19 July 2017 (UTC)