- Show that each component of X is an open subset of X.
- Show that, if X is connected, then X is path-connected.
is bijective and continuous, show that f is a homeomorphism.
be the stereographic projections. Write down explicit
formulas for these maps, and prove that they are homeomorphisms.
be the unit n-disk,
with boundary 
. Prove that
is homeomorphic to 
.
on
by writing
if and only if
. The quotient space
is called the (real) projective n-space.
Let 
,
be the canonical
projection. For each i, 
,
define a subset 
of
by
Prove the following facts, which together show that
is an n-manifold.
-
is open in
.
-
covers 
.
- There is a homeomorphism
.
-
is compact, connected, and Hausdorff.
-
.
-
is an
-manifold.
-
is an n-manifold.
.
for V and
for W defines an isomorphism
of 
with the real vector space of
matrices with real coefficients, 
.
Shows that this isomorphism transforms composition
into matrix multiplication
alexsuciu@neu.edu