|This article was nominated for merging with Homotopy extension property on 12 December 2015. The result of the discussion was no consensus.|
|WikiProject Mathematics||(Rated Start-class, Low-importance)|
I think A must be a subset of X.
- Well then i is simply an inclusion map. It is better in my opinion to consider a mapping cylinder of A ajoined to X instead of just (X x 0) U (A x I).
In the third basic theorem, I don't think the biconditional always holds. I think A may have to be closed for the reverse direction to hold.
A is always closed. Follows from retraction or extension property. We may need Hausdorff though, but I am pretty certain we do not. —Preceding unsigned comment added by 22.214.171.124 (talk) 03:26, 2 October 2007 (UTC)
I added the compactly generated condition for a cofibration to be closed inclusion. May's book page 42 seems to say this is so, but maybe he means an arbitrary space. I haven't found a proof yet but I'll get back if I do. Money is tight (talk) 15:19, 24 January 2011 (UTC)
- May prefaces chapter 6 with a reminder that all spaces are to be taken as compactly generated. i.e. A, X, Y. Exactly how this definition might be altered if they are not is not stated. User:Linas (talk) 01:44, 16 August 2013 (UTC)
The "definition" given in the text (closed inclusion = injective with closed image) is misleading. The subspace that is included also needs to carry the subspace topology! I suggest writing "inclusion of a closed subspace" instead of "closed inclusion" to avoid this ambiguity. --126.96.36.199 (talk) 08:01, 6 May 2015 (UTC)
Indeed, the current presentation is highly misleading (this isn't isolated to Wikipedia). For example, the obvious map from [0,1) is injective with closed image, but it clearly doesn't satisfy the HEP. Cofibrations should be certain types of maps, not pairs of a space and subspace (I think). If you're going to mix the two, then clear and careful explanation should be given. But the treatment on Wikipedia at the moment is to define cofibrations as certain kind of maps satisfying the HEP, which on the corresponding page is a property of pairs of spaces. 2A02:C7D:BC32:5200:2C09:F07C:D9B0:14F2 (talk) 17:34, 18 July 2017 (UTC)