Singularities from topological viewpoint

Question

I intuitively understand that \begin{align} f:(-\infty.0)\cup(0,+\infty) &\to \mathbb{R} \\ x &\mapsto \dfrac{1}{x} \end{align} is "singular" at $x=0$, however, is there a topological description of singularities, in terms of say open sets? Or does one need to discuss singularities in the context of differentiable manifolds?