Simplicial Structures in Topology

I.1 Topology 23
(I.1.38) Lemma. Let X and Y be topological spaces where X is compact 3
Hausdorff. Then the function
ε : X × M(X,Y) → Y
(evaluation function), defined by ε(x, f )= f (x) for every x ∈ X and every f ∈
M(X,Y ), is continuous.
Proof. Take arbitrarily (x, f ) ∈ X × M(X,Y ) andanopensetU ⊂ Y with f (x) ∈ U.
Since X is compact Hausdorff and f −1 (U) is open in X, there exists an open set V
of X such that
x ∈ V ⊂ V ⊂ f −1 (U)
where V is compact (see Corollary (I.1.35)). We end the proof by noting that (x, f ) ∈
V ×W V,U and ε(V ×W V,U ) ⊂ U. �
(I.1.39) Theorem. Let X be a compact Hausdorff space; then, if ˆf : Z → M(X,Y )
is continuous, so is f : X × Z → Y.
Proof. It is enough to note that f is the composite of
1X × ˆf : X × Z → X × M(X,Y) and ε : X × M(X,Y) → Y
and apply Lemma (I.1.38). �
(I.1.40) Corollary. Let q: X −→ Y be a quotient map. If Z is a compact Hausdorff
space, then also
q × 1Z : X × Z −→ Y × Z
is a quotient map.
Proof. The quotient map q: X → Y has the following property which characterizes
the quotient topology on Y : a function g: Y → W is continuous if and only
if gq: X → W is continuous (see Exercise 7 at the end of this section). We must
therefore prove that, for every topological space W and every g: Y × Z → W , g
is continuous if and only if h = g(q × 1Z) is continuous: If g is continuous, h is
undoubtedly continuous. Now, the following commutative diagram
h
X × Z ��
��
W
��
��
q × 1Z ��
��
��
g
��
��
Y × Z
3 Actually it is not necessary to ask that X be compact; in fact, it is sufficient to request that X be
locally compact, that is to say, that every point of X has a compact neighbourhood.