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Let $X$ be a topological space, locally compact, hausdorff, and maybe not compact.

Consider a compactification of $X$, $(Y,\varphi)$

What hypothesis would imply that the residue $Y\backslash\varphi(X)$ is connected?

Camilo Andrés Acevedo Ardila
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  • It would rely heavily on the specific compactification procedure, roughly speaking "the collection of different ways to run to infinity". In an extreme example, the residue of the one-point compactification is obviously connected. In a general setting where $Y$ is any compact space with $X$ densely embedded, there isn't a lot to say. – Lieven May 16 '24 at 16:53
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    The remainder of any $T_2$-compactification is a continuous image of the remainder of the Stone-Cech compactification. Hence, each remainder is connected, iff the remainder of the Stone-Cech compactification is connected. For instance, this is the case for $[0,1)$, but not for $(0,1)$. – Ulli May 16 '24 at 17:32
  • @Ulli Is there any book or article where I can find this result you are talking about? It would help me really much, thanks! – Camilo Andrés Acevedo Ardila May 16 '24 at 19:34
  • @Camilo Andrés Acevedo Ardila: which one do you exactly asking for? – Ulli May 16 '24 at 19:36
  • @Ulli Actually both results you mentioned, the one about the remainder of any T2 compactification, and the iff statement you mentioned. – Camilo Andrés Acevedo Ardila May 16 '24 at 19:38
  • For the first one see here: https://math.stackexchange.com/questions/426726/remainders-of-compactifications-are-images-of-the-stone-%c4%8cech-remainder – Ulli May 16 '24 at 19:44
  • The iff-statement then should be clear, since continuous images of connected spaces are connected. Please let me know, what is unclear. – Ulli May 16 '24 at 19:47
  • @Ulli : you are right, I don't know why I had this false view in mind : connectedness and path-connectedness are topological properties. – Jean Marie May 16 '24 at 19:50
  • @Ulli thanks, it's clear now, and would there maybe be something that implies that the remainder of the Stone-Cech compactification is connected? – Camilo Andrés Acevedo Ardila May 16 '24 at 20:22
  • @Camilo Andrés Acevedo Ardila For instance, what I called compactly connected in my answer here. But I think the proof is too involved for a comment. – Ulli May 16 '24 at 20:54
  • This should also be detectable in the end compactification (see https://en.wikipedia.org/wiki/End_(topology)) – ronno May 17 '24 at 13:41
  • I think having a unique end should be more or less the same as compactly connected as in @Ulli's last comment – ronno May 17 '24 at 13:43
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    @ronno: "compactly connected" is more general: for instance, spaces having a unique end are $\sigma$-compact, if I understood the Wikipedia definition correctly. For instance, the long line is locally compact, compactly connected, but not $\sigma$-compact, hence has not a unique end. – Ulli May 17 '24 at 15:12

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