Homogeneity of powers of spaces and the character

Arhangel’skii, A.

doi:10.1090/S0002-9939-05-07774-9

Homogeneity of powers of spaces and the character
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by A. V. Arhangel’skii PDF

Proc. Amer. Math. Soc. 133 (2005), 2165-2172 Request permission

Abstract:

A space is said to be power-homogeneous if some power of it is homogeneous. We prove that if a Hausdorff space $X$ of point-countable type is power-homogeneous, then, for every infinite cardinal $\tau$, the set of points at which $X$ has a base of cardinality not greater than $\tau$, is closed in $X$. Every power-homogeneous linearly ordered topological space also has this property. Further, if a linearly ordered space $X$ of point-countable type is power-homogeneous, then $X$ is first countable.

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