TITLE: A Compact Hausdorff Space with no Manifold Structure?
QUESTION [5 upvotes]: What is an example of a compact Hausdorff space that cannot be given the structure of a 
(i)   differential manifold
(ii)  topological manifold?

REPLY [3 votes]: The interval $[0,1]$ is a compact Hausdorff space which doesn't carry the structure of a manifold without boundary. (Of course, it carries the structure of a manifold with boundary.)