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1. It is easily seen that every compact ordered space with infinitely many points which has a countable base admits continuously many autohomeomorphisms.For, if there are countably many isolated points, the assertion is obvious. In the other case the assertion follows from the fact that there is either a separable connected subspace which is consequently homeomorphic to an interval of the real numbers or the space is zero-dimensional and so is homeomorphic to the Cantor set (possibly except for finitely many isolated points).

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