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I know that Alexandroff compatification is unique, and if the Alexandroff compatification of two spaces are not homeomorphic, then the spaces can't be. Does uniqueness stand in n point compatifications? And what does homeomorphism (or not) between the compatifications tells us about the original spaces? Finally, if A has a 2 points compatification, and B doesn't (maybe has 1 point compatification) can we say A and B are not homeomorphic?

Temitope.A
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  • Every space has a 1-point compactification. – Cameron Buie Jul 17 '12 at 15:12
  • If the property $P_k $ "has a k-point compactification" is defined in terms of topology, then of course it is a topological invariant: two homeomorphic spaces will either both have $P_k$ or not have $P_k$. Could you define $P_k$ for us, by the way? –  Jul 17 '12 at 15:18
  • @Cameron I think the space has to be locally compact Hausdorff, see Wikipedia. (Compactifications are usually required to be $T_2$.) – Martin Sleziak Jul 17 '12 at 15:21
  • @CameronBuie I don't think so..... –  Jul 17 '12 at 15:23
  • @LeonidKovalev We Know that the topological space Y is a compactification of the topological space X, if the space Y is compact and hausdorff and X is dense in Y. If for a positive integer n we have a compactification Y that Y−X has only n elements, we say that Y is the n point compactification of X. Now I'm not sure about the topological behaviour of this property because I don't know anything about uniqness; but maybe I'm just confused. I can't find any complete reference online on n-points compatification. And I can't fine anything in Kosniowski or Dugundji – Temitope.A Jul 17 '12 at 15:36
  • For a related mathoverflow question, see A question about some special compactifications of $\mathbb R$. – Dave L. Renfro Jul 17 '12 at 15:53
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    @Martin: You can see here what I mean by 1-point compactification (apologies for the belatedness of the clarification). I also mention there that for the compactification to be Hausdorff, the original space must be locally compact Hausdorff. – Cameron Buie Jun 13 '13 at 04:13
  • @BenjaLim: See the above comment for (much belated) clarification. – Cameron Buie Jun 13 '13 at 04:13

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Uniqueness does not hold: for example, $(0,1)\cup (2,3)$ can be 2-point compactified into a circle (by adding $0=3$ and $1=2$) or into two disjoint circles (by adding $0=1$ and $2=3$).

However this is of no consequence for topological invariance. As long as a property is defined in terms of topology on $X$, it is invariant under homeomorphism. If you wish to make this more precise, you can rephrase the definition:

$X$ having an $n$-point compactification means that $X$ is homeomorphic to some space of the form $Y\setminus \{y_1,\dots,y_n\}$ where $Y$ is compact Hausdorff and $y_i\in Y$ are distinct.