Prove that the Sphere with a Hair in $\mathbb{R} ^{3}$ is not Locally Euclidean at q, hence it can not be a Topological Manifold.

A connected manifold has a unique dimension $n$, and every point of $X$ then has an open neighbourhood homeomorphic to the open unit ball $\mathbb D^n\subset \mathbb R^n$.

However in the pictured $X$ the points different from $q$ on the hair have an open neigbourhood homeomorphic to $\mathbb D^1$ , whereas the points different from $q$ on the sphere have an open neigbourhood homeomorphic to $\mathbb D^2$.

Since $X$ is connected this proves that it is not a manifold, since it cannot have a unique dimension.