We will say that a Hausdorff topological space $X$ is a smooth manifold if there is an open cover $(U_{\alpha})$ of $X$ and a corresponding collection of homeomorphisms $\varphi_{\alpha} : U_{\alpha} \to V_{\alpha} \subset \mathbb{R}^n$ such that on any overlap $U_{\alpha} \cap U_{\beta}$, the maps $$\varphi_{\beta} \circ \varphi_{\alpha}^{-1} : \varphi_{\alpha}(U_{\alpha} \cap U_{\beta}) \longrightarrow \varphi_{\beta}(U_{\alpha} \cap U_{\beta})$$ are smooth.

Note that I have omitted the assumption of paracompactness. I recall vaguely from my undergraduate years of my lecturer telling us that there are good reasons for omitting paracompactness from the definition of a manifold unless one wants to look at metric properties of $X$ or integrate.

His expertise was in complex geometry and homogenous spaces. I vaguely recall his justification for this being that every group can be declared a Lie group if one allows the more general definition since one is permitted to have an uncountable number of connected components.

I am not sure if I have correctly remembered this, but I was hoping someone could maybe elaborate on this thought further?

wild. $\endgroup$ – Benoît Kloeckner Nov 25 '19 at 11:5011more comments