Simplicial Structures in Topology

III.5 Real Projective Spaces 129
We ask the reader to prove that
S n |Z2 = RP n ∼ = (R n+1 � (0,...,0))/ ≡ .
In this way, we adopt the definition RP n =(R n+1 � (0,...,0))/ ≡.
We now prove that RP n+1 may be obtained from RP n by “adjunction” of an
(n+1)-dimensional disk D n+1 ; more precisely, RP n+1 is the pushout of the diagram
ın
S n
��
D n+1
qn ��
RP n
where ın is the inclusion of S n in D n+1 and qn is the map that takes a point
(x0,...,xn) ∈ S n into its equivalence class
We start by constructing the pushout
[(x0,...,xn)] ∈ (R n+1 � (0,...,0))/ ≡ .
S n
��
ın
D n+1
qn ��
qn ¯
and the following commutative diagram
where
and
ın
S n
��
D n+1
gn(x0,...,xn)=
�
RP n
ın ¯
��
��
n
RP ⊔qn Dn+1
qn ��
gn
(x0,...,xn,
RP n
jn
��
��
n+1
RP
�
1 −
n
∑
0
jn([(x0,...,xn)] = [(x0,...,xn,0)]
|xi| 2 )
for every (x0,...,xn) ∈ S n and [(x0,...,xn)] ∈ RP n . By the definition of pushout,
there is a unique continuous function
ℓ: RP n ⊔qn Dn+1 → RP n+1
�