Orientable double cover

Orientable double cover

A closely related notion uses the idea of covering space. For a connected manifold ${\displaystyle M}$ take ${\displaystyle M^{*}}$, the set of pairs ${\displaystyle (x,o)}$ where ${\displaystyle x}$ is a point of ${\displaystyle M}$ and ${\displaystyle o}$ is an orientation at ${\displaystyle x}$; here we assume ${\displaystyle M}$ is either smooth so we can choose an orientation on the tangent space at a point or we use singular homology to define orientation. Then for every open, oriented subset of ${\displaystyle M}$ we consider the corresponding set of pairs and define that to be an open set of ${\displaystyle M^{*}}$. This gives ${\displaystyle M^{*}}$ a topology and the projection sending ${\displaystyle (x,o)}$ to ${\displaystyle x}$ is then a 2-to-1 covering map. This covering space is called the orientable double cover, as it is orientable. ${\displaystyle M^{*}}$ is connected if and only if ${\displaystyle M}$ is not orientable.