Homogeneous countably compact spaces $X$ and $Y$ whose product $X\times Y$ is not pseudocompact are constructed. It is proved that all compact subsets of homogeneous subspaces of the third power of an extremally disconnected space are finite. Moreover, under CH, all compact subsets of homogeneous subspaces of any finite power of an extremally disconnected space are finite and all compact subsets of homogeneous subspaces of the countable power of an extremally disconnected space are metrizable. It is also proved that all compact homogeneous subspaces of finite powers of an extremally disconnected space are finite, which strengthens Frol\'{\i}k's theorem.

中文翻译：

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极不连通空间乘积的齐次子空间

构造齐次可数紧致空间 $X$ 和 $Y$，其乘积 $X\times Y$ 不是伪紧致。证明了一个极不连通空间的三次方齐次子空间的所有紧子集都是有限的。此外，在CH下，极不连通空间的任何有限幂的齐次子空间的所有紧子集都是有限的，并且极不连通空间的可数幂的齐次子空间的所有紧子集都是可度量的。还证明了极不连通空间的有限幂的所有紧齐齐子空间都是有限的，这加强了 Frol\'{\i}k's theorem。