The product structure of homogeneous spaces (Q910751)

We prove that a homogeneous locally compact separable metrizable space can be written as the product of two spaces, one of which is connected and the other of which is totally disconnected. We arrived at this result while showing that certain one-dimensional spaces can be endowed with the structure of a flow in a non-trivial way. \textit{M. W. Mislove} and \textit{J. T. Rogers} implicitly proved the compact case of our theorem, that every compact homogeneous metric space is a product. We also present an example of a homogeneous separable metric space which does not have a product structure. Hence the assumption of local compactness in our theorem is essential. As a corollary to our main result we get that in a locally compact homogeneous separable metrizable space the component of a point coincides with the quasi-component of that point.