Simplicial Structures in Topology

40 I Fundamental Concepts
(I.3.1) Lemma. Let φ : G × X → X be an action of a topological group G on a
topological space X. For every g ∈ G, the function
is a homeomorphism.
φ(g): X → X, x ↦→ xg
Proof. The function φ(g−1 ) is the inverse of φ(g). The continuity of φ(g) is immediate;
in fact, φ(g) is the composite of the action φ and the inclusion X ×{g} ↩→
X × G. �
An action φ : X ×G → X gives rise to an equivalence relation ≡φ in X (a partition
of X into G-orbits):
x ≡φ x ′ ⇐⇒ (∃g ∈ G)x ′ = xg.
The equivalence class [x] of the element x ∈ X is the orbit of x, also denoted by
xG (or Gx, if the action is on the left); and we write X/G to indicate the set X/≡φ
of the orbits of X. ThesetX/Gwiththequotient topology given by the canonical
epimorphism
q: X → X/G , x ↦→ [x]
is called orbit space of X under the action of G.
(I.3.2) Lemma. The quotient map q: X → X/G is open. If G is a finite group, then
qisalsoclosed. 5
Proof. Let U be any open set in X. Then
q −1 (q(U)) = {x ∈ X | q(x) ∈ q(U)}
= {x ∈ X | (∃ g ∈ G)(∃ y ∈ U) x = yg}
= {x ∈ X | (∃ g ∈ G)x ∈ Ug}
= �
φ(g)(U).
g∈G
Since, according to Lemma (I.3.1), the functions φ(g) are homeomorphisms for
every g ∈ G, q −1 (q(U)) is a union of open sets in X; hence q(U) is open in X/G.
If G is finite and K ⊂ X is closed,
q −1 (q(K)) = ∪g∈Gφ(g)(K)
is closed as a finite union of closed sets. �
We recall that the real projective plane RP2 in Example (I.1.5) on p. 5 was
constructed as the quotient space D2 ≡ , obtained from the unit disk D2 by identifying
(x,y) =(−x,−y) for every (x,y) ∈ D2 such that x2 + y2 = 1. Then, we
note that the discrete topological group Z2 = {1,−1} acts on the unit sphere
5 That is to say, it takes closed sets into closed sets.