A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ \mathbb{R} ^{n}$ and $V ⊂ \mathbb{R} ^{m}$ are homeomorphic, then $n = m$. Prove that the sphere with a hair in $\mathbb{R} ^{3}$ is not locally Euclidean at $q$. Hence it cannot be a topological manifold.
I am new to the theory of manifolds, so I have no idea. 
Best Answer
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.