Real projective n-space
Real projective space, is defined to be the space of all lines through the origin in . Each such kind is determined by a non-zero vector in , unique up to scalar multiplication, and is topologized as the quotient space of under the equivalence relation for scalars . We can restrict to vectors of length 1, so is also the quotient spaces , the sphere with antipodal points identified. This is equivalent to saying that is the quotient space of a hemisphere with antipodal points of identified. Since with antipodal points identified is just , we see that is obtained from by attaching an -cell, with the quotient projection as the attaching map. It follows by induction on that has a cell complex structure with one cell in each dimension . Since is obtained from by attaching an -cell, the infinite union becomes a Cell Complex with one cell in each dimension. We can view as the space of lines through the origin in .
Complex Projective n-space
Complex projective n-space is space of complex lines through the origin in , that is, 1-dimensional vector subspaces of . As in the case of Real projective n-space, each line is determined by a non-zero vector in, unique up to scalar multiplication, and is topologized as the quotient space of under the equivalence relation for .
Equivalently, this is the quotient of the unit Sphere with for .
It is also possible to obtain as a quotient space of disk under the identifications for , in the following way:
The vectors in with last coordinate real and nonnegative are precisely the vectors of the form with . Such vectors form the of the function . This is a disk bounded by the sphere consisting of vectors with . Each vector in is equivalent under the identifications to a vector in , and the latter vector is unique if its last coordinate is non-zero. If the last coordinate is zero, we have just the identifications for .
From the description of as the quotient of under the identifications for , it follows that is obtained from by attaching a cell via the quotient map . So by induction on , we obtain a cell structure
with cells only in even dimensions. Similarly, has a cell structure with one cell in each even dimension.
How to Draw a Cell Complex (or CW Complex)
Let we try to construct a space by following procedure:
- Start with a discrete set , whose points are regarded as -cells.
- Inductively, form the -skeleton from by attaching -cells via maps . This means that is the quotient space of the disjoint union of with a collection of n-disks under the identifications for . Thus as a set
where each is an open disk.
- One can either stop this inductive process at a finite stage, setting for some , one can continue indefinitely, setting .
A space constructed in this way is called a CELL COMPLEX or CW COMPLEX.