4-manifold

Remarks

Example: the $4$-sphere $S^4 = \set{x \in \R^5 : \abs{x} = 1}$ is Hausdorff and second countable as a subspace of $\R^5$, and stereographic projection from a point $p$ maps $S^4 \setminus \set{p}$ homeomorphically onto $\R^4$, which gives clause 3.

Non-example: take two copies of $\R^4$ and glue them along the complement of the origin. Every point has a neighborhood homeomorphic to $\R^4$, but the two origins have no disjoint neighborhoods, so clause 1 fails.

By invariance of domain, no nonempty open subset of $\R^4$ is homeomorphic to an open subset of $\R^n$ with $n \ne 4$, so the dimension is well defined. The $E_8$ manifold is a closed $4$-manifold with no smooth structure.