Kottler-Moller coordinates, metric "covers" a region

[Gene Naden]

Working through an online course "Introduction to General Relativity." They give the metric for, Kottler-Moller coordinates, i.e. $$ds^2=(1+ah)^2d\tau^2-dh^2-dy^2$$
and say that it "covers" the Rindler wedge in flat space time, which is defined by $$0<x<\infty,-x<t<x$$
I am having difficulty seeing precisely what it means for a metric to "cover" a region. Why doesn't the metric cover all of space-time? What is special about $$x=0$$ or $$t=0$$? Is there a singularity or degeneracy in the metric for some points?

 

[Orodruin]

A metric does not cover a region of spacetime. However, a coordinate chart does. The metric clearly has a singularity at $h = -1/a$, which is a coordinate singularity that can be removed by going to, e.g., standard Minkowski coordinates

 

[Gene Naden]

Thank you