# Example of a CW complex not homeomorphic to the realization of a simplicial set?

I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't isomorphic to the total singular complex of a CGWH space. Are there?) Would someone mind providing an example of one (and an example for the opposite statement as well, if it is true)?

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Heh, I just realized that we could call the "opposite statement" the "adjoint statement". –  Harry Gindi Aug 5 '10 at 20:13

The mapping cylinder of a really messy continuous map $I\to I$