Let $M$ be a connected, compact, boundaryless topological manifold of dimension $n$. Let $X$ be the space obtained by removing some point from $M$.
My question is, does $X$ admit a deformation-retract onto a CW complex of dimension $n-1$?
I can see directly why this is true for $n=2$, i.e. for $M$ a surface; then we know $M$ can be obtained by taking some plane polygon and identifying pairs of sides; so if we remove a point from the interior, this deformation-retracts to the 1-dimensional CW complex obtained by identifying the pairs of edges (as the sides in the original polygon) and discarding the interior of the polygon.
However, I'm not sure how to generalize this argument to $n>2$. Would anyone know how to proceed in this case?