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Does the Sorgenfrey line have the homotopy type of a CW-complex? I know that the Sorgenfrey line is a paracompact, Hausdorff space, but cannot be a manifold because this space is not locally compact. How about a CW-complex?

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The Sorgenfrey line is totally disconnected, while CW complexes have non-trivial connected components, except for zero-dimensional CW-complexes, which are discrete.

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