Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don't come along with this, like a global Hausdorff condition mentioned here, second countability or paracompactness etc.
Mainly it would seem to rule out certain pathological examples, or to simplify proofs, rather than saying 'Let $M$ be a Hausdorff, Second countable, Manifold$\ldots$' every time.
From algebraic topology the existence of a universal covering space of a topological space $X$, required $X$ to be connected, locally path connected, and semi locally simply connected. In the course I did we said path connected, locally path connected and semi locally simply connected. I believe these are equivalent.
My question is: For a $M$ a manifold, does $M$ satisfy the existence criterion?
Or should I specifically require that $M$ is a connected manifold, and it would seem that locally path connected and semi locally simply connected come from the charts, or local homeomorphisms to $\Bbb R^n$.