# When is the one-point compactification well-pointed?

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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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). –  Ricardo Andrade Jul 31 '13 at 22:51
@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? –  John Klein Aug 1 '13 at 0:19
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. –  Ricardo Andrade Aug 1 '13 at 9:28
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. –  John Klein Aug 1 '13 at 23:21
@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. –  Ricardo Andrade Aug 2 '13 at 2:06