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$\beta \mathbb{N}$; retracts; two to one map; Stone-Čech compactification
Eric van Douwen produced in 1993 a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of $\beta {\mathbb{N}}$. We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of $\beta {\mathbb{N}}$ expanding on earlier work of Balcar and Błaszczyk (1990) and Simon (1987).
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