A Compact Hausdorff Space with no Manifold Structure?

Question

What is an example of a compact Hausdorff space that cannot be given the structure of a

(i) differential manifold

(ii) topological manifold?

Answer 1

The union of the $x$-axis and $y$-axis in $\mathbb{R}^2$, intersected with the closed unit disk in $\mathbb{R}^2$.

Answer 2

The Cantor set is a nice example.

Answer 3

Any set that has an accumulation point and an isolated point

Answer 4

The cartesian product $[0,1]^{\Bbb R}$ is obviously compact and Hausdroff, but isn't locally homeomorphic to any $\Bbb R^n$. The really interesting (and hard) case is a topological manifold without differentiable structure.

Answer 5

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.)