-1
$\begingroup$

Is the following statement true or not?

A locally compact Hausdorff space $X$ is a group if and only if its Stone–Čech compactification$\beta X$ is a group.

Thanks.

  • 0
    $\beta\mathbb{Z}$ is not a group, I believe, while $\mathbb{Z}$ is...2012-12-22
  • 0
    In general (see http://dutiaw37.twi.tudelft.nl/~kp/onderwijs/topologie/d17-betaX.pdf, e.g.) we have that a pseudocompact (Tychonoff) topological group is such that its group operations can be extended to its Cech-Stone compactification.2012-12-22
  • 2
    You should explain what "is a group" means.2012-12-22
  • 0
    There is a way to compactify topological groups (http://en.wikipedia.org/wiki/Bohr_compactification) but the underlying topological space of the Bohr compactification isn't the Stone-Cech compactification. For example, the Bohr compactification of $\mathbb{Z}$ is the Pontrjagin dual of $S^1$ with the discrete topology...2012-12-22
  • 0
    Here a group means that it is a topological group under the given topology.2012-12-23

1 Answers 1

6

I assume both occurrences of "is a group" mean "is the underlying space of a topological group." Then Henno Brandsma's first comment gives a counterexample, because $\beta\mathbb Z$ is indeed not the underlying space of a topological group. The reason is that it is not homogeneous: The points in $\mathbb Z$ are isolated and the others are not.

  • 0
    Thanks for answering my question. Bohr compactification is just for groups. I am wondering that given a locally compact Hausdorff space $X$, is there any compactification of $X$, say $Y$, such that $X$ is a topological group if and only if $Y$ is a compact group?2012-12-23