What is a singular space?

Question

A book I am reading on orbifolds uses the word singular space but doesn't say what it means. The book is Orbifolds and Stringy Topology by ALR the quote is "Orbifolds are singular spaces that are locally modelled on a quotient of a smooth manfiold by the action of a finite group".

Answer 1

A manifold is locally modeled on $\Bbb R^n$. An orbifold is locally modeled on $\Bbb R^n/G$, where $G$ is a finite subgroup of $O(n)$.

On the one hand, you can think of this as allowing for singular models of manifolds. For $\Bbb Z/n \subset SO(2)$, $\Bbb R^2/\Bbb Z/n$ is a cone with cone angle $2\pi/n$; or for $\Bbb Z/2 \subset O(n)$ (where the nontrivial element acts by negation on $\Bbb R^n$), the resulting object is homeomorphic to an open cone on $\Bbb{RP}^{n-1}$.

On the other hand, note that the resulting object is more structured than a topological space (or even a smooth manifold; your charts vary not just over open subsets of $\Bbb R^n$, but rather that equipped with an orthogonal group action). For instance, the cone example above is topologically equivalent to $\Bbb R^2$, but not equivalent as an orbifold. So the visualization you should have of an orbifold should really look like a quotient somehow when you visualize it.