The Dimension of a Manifold One thing that may seem clear at
first blush about manifolds actually takes a little thought to
be sure about. Specifically, our definition says that each point
is contained in some coordinate patch , which comes with a
coordinate map . But does the here have to be the same for all
points ? Well, let’s start by considering what happens if two
coordinate patches intersect each other. Let and be open subsets
with not empty, and with coordinate maps and . Now we can
restrict our coordinate patches to the intersection, getting and
. These are two different coordinate patches with two different
coordinate maps on the same open subset . So what? So now we can
use in reverse to lift up from into , and then use forward to
drop down into . This composition is thus a homeomorphism from
an open region in to an open region in . And this is absolutely
impossible unless . So now we know that any two coordinate
patches that intersect must use the same value of . Does this
mean that we always have