5.6. Groupoids 201(although there are important examples of non-Hausdorff locally compact
groupoids). By the Hausdorff property, G(O) is closed.
Lemma 5.6.2. Let G be an etale groupoid. Then, Q(O) c G is a clopen
subset.Proof. To show that Q(O) is open in G, let x E Q(O) be given and take an
open neighborhood U of x in G such that s is homeomorphic on U. Since
Q(O) n U is an open subset of Q(O), V = U n s-^1 ( Q(O) n U) is an open
neighborhood of x in G. We claim that V c Q(O). Indeed, for every 'Y E V,
we have s('Y) E U. Since s is injective on U, s('Y) = s(s('Y)) implies that
'Y = s('Y)· D
Example 5.6.3 (Groups, spaces and actions on spaces). Let r be a group.
Then Q(O) = { e} c G = r is a groupoid. Let X be a locally compact space.·
Then, Q(O) = G =Xis a groupoid. Now, let r act on X. Then,
X ><I I'= {(x,s,y) EX XI' XX: x = s.y}
is a groupoid with s(x,s,y) = (y,e,y), r(x,s,y) = (x,e,x) and, finally,
(x, s, y)(y, t, z) = (x, st, z). The third component is often omitted because
y is uniquely determined by (x, s). The topology on X ><1 r is nothing but
that coming from X x r, where r is viewed as a discrete space. With
this topology, x ><I r is an etale locally compact groupoid, which is called
a transformation groupoid. The groupoid X ><1 r is a special case of the
groupoid semidirect product.
Definition 5.6.4. A partial homeomorphism on X is a homeomorphism
f: U ---+ V between open subsets of X.^15 We denote by dom(f) = U the
domain of f. For partial homeomorphisms f : U ---+ V and g : V' ---+ W, we
simply write gof for (glvnV' )o(JIJ-l(V'))· We note that fldom(f)nU = f oidu.
A pseudogroup g on X is a collection of partial homeomorphisms with the
following properties:
(1) (units) the identity idu on every open subset UC X is in Q;
(2) (composition) if f, g E Q, then go f E Q;
(3) (inverse) if f: U---+ Vis in Q, then 1-^1 : V---+ U is in Q.
Sometimes in the literature the following is also required:
(4) (extension) if f is a partial homeomorphism with an open covering
{Ui}i of dom(f) such that f o idui E Q, then f E Q.
Example 5.6.5. If X is a smooth (or Riemannian, etc.) manifold, then
the collection of all partial diffeomorphisms (or partial isometries, etc.) is
15we allow th~ empty map which is the unique partial homeomorphism between empty
subsets. ·