M6140 Topology Exercises - 8th Week (2020)
1 Mapping Spaces
Definition 1. By C(X, Y ) we denote the set of all continuous maps from a topological space X to a
topological space Y .
Exercise 1. Suppose that X is a non-empty topological space and Y is a topological space. Show
that C(X, Y ) is a singleton iff Y is a singleton.
Exercise 2. Suppose that X is a non-empty topological space and Y is a topological space. Prove
that the cardinality of Y is less or equal to the cardinality of C(X, Y ).
Definition 2. Let A ⊆ X, B ⊆ Y be subsets in topological spaces X, Y . Define
W(A, B) := {f ∈ C(X, Y ) | f(A) ⊆ B}.
Exercise 3. Suppose that X is a non-empty topological space and Y is a topological space. Prove
that
S pw
:= {W(a, U) | a ∈ X, U is an open set in Y }
is a cover of C(X, Y ).
Definition 3. The set S pw can be viewed as a subbase, the generated topology is called the topology
of pointwise convergence, and the resulting topological space is denoted by Cpw(X, Y ).
Exercise 4. Suppose that X is a non-empty topological space and Y is a topological space. Prove
that
S co
:= {W(K, U) | K is compact in X, U is an open set in Y }
is a cover of C(X, Y ).
Definition 4. The set S co can be viewed as a subbase, the generated topology is called the compactopen
topology, and the resulting topological space is denoted simply by C(X, Y ).
Exercise 5. Prove that T (Cpw(X, Y )) ⊆ T (C(X, Y )) for each pair of topological spaces X, Y , where
X is non-empty.
Exercise 6. Show that T (Cpw(I, I)) = T (C(I, I)).
Exercise 7. Suppose that X is a non-empty topological space and Y is a T2 topological space. Prove
that Cpw(X, Y ) and C(X, Y ) are both T2.
Exercise 8. Suppose that X, Y, X , Y are topological spaces and g: X → X, h: Y → Y are continuous
maps. Show that ϕ: C(X, Y ) → C(X , Y ) given by ϕ(f) := h ◦ f ◦ g is continuous.
Exercise 9. Prove that Cpw(I, I), C(I, I) aren’t homeomorphic.
Exercise 10. Suppose that X is a T2 space, Y is a T2 locally compact space, and Z is a topological
space. Show that there exists a homeomorphism C(X × Y, Z) ∼= C(X, C(Y, Z)).
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