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This is a follow up to my previous question.

Question:

Is there a reasonably natural set of conditions which guarantee that the one-point compactification $X^+$ of a locally compact Hausdorff space $X$ is well-pointed?

(Where, of course, the basepoint I'm referring to is the point at $\infty$.)

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    $\begingroup$ I added the tag 'gn.general-topology'. Feel free to remove it if you don't think it applies. The tag 'gt.geometric-topology' may also be relevant (but may not be directly applicable). $\endgroup$
    – Ricardo Andrade
    Commented Jul 31, 2013 at 22:51
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    $\begingroup$ @ViditNanda: I looked at the paper you suggested, but I am not a shape theorist, so I couldn't make any sense of it. The only statement I could understand was the acknowledgement: "We completed this work during compulsory military service at Military High School in Belgrade. We thank superior officers for providing conditions stimulating research." Could you please tell me where I would find the answer to my question in this paper? $\endgroup$
    – John Klein
    Commented Aug 1, 2013 at 0:19
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    $\begingroup$ Here is an example which I find counter-intuitive, and may be interesting to keep in mind. The Whitehead manifold is a space with a highly non-trivial topology at infinity whose one-point compactification is nevertheless well-pointed. See this answer by Sergey Melikhov for an explanation. $\endgroup$
    – Ricardo Andrade
    Commented Aug 1, 2013 at 9:28
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    $\begingroup$ I just looked again at Hughes & Ranicki's book. It seems that they give an answer in their setting: Prop 7.11 states that a "forward tame" ANR $W$ has the homotopy type of a based CW complex. $\endgroup$
    – John Klein
    Commented Aug 1, 2013 at 23:21
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    $\begingroup$ @John: Forward tameness appears to be a useful condition. Nevertheless, I was hoping for a more general criterion. For example, the Whitehead manifold I mentioned above is not forward tame. $\endgroup$
    – Ricardo Andrade
    Commented Aug 2, 2013 at 2:06

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