Page 247 - DMTH503_TOPOLOGY

Unit 29: Pointwise and Compact Convergence

Proof: Given a point (X, f) of X × e (X, Y) and an open set V in Y about the image point e (x, f) = Notes
f (x), we wish to find an open set about (x, f) that e maps into V. First, using the continuity of f and
the fact that X is locally compact Hausdorff, we can choose an open set about x having compact
closure , such that f carries into V. Then consider the open set × S ( , V) in X × e (X, Y). It
is an open set containing (x, f). And if (x, f) belongs to this set, then e (x, f) = f(x) belongs to V,
as defined.
Theorem 4: Let X and Y be spaces, give e (X, Y) the compact-open topology. If f : X × Z  Y is
continuous, then so is the induced function F : Z  e (X, Y). The coverse holds if X is locally
compact Hausdorff.
Proof: Suppose first that F is continuous and that X is locally compact Hausdorff. It follows that
f is continuous, since f equals the composite.
x i 
e
F
X × Y  X × e (X × Y)  Y,
where i is the identity map of X.
x
Now suppose that f is continuous. To prove continuity of F, we take a point Z of Z and a sub-basic
0
element S (e, ) for C (X, Y) containing F (Z ) and find a neighborhood W of Z that is mapped by
0 0
F into S (C, ). This will suffice.
The stalement that F (Z ) lies in S (C, ) means simply that (F (Z )) (x) = f (x, Z ) is in for all
0 0 0
x  C. That is, f (C × Z )  . Continuity of f implies that f ( ) is an open set in X × Z containing
–1
0
C × Z . Then
0
f –1 ( ) (C × Z)
is an open set in the subspace C × Z containing the slice C × Z .
0
The tube lemma implies that there is a neighborhood W of Z in Z such that the entire tube C × W
0
–1
lies in f ( ). Then for Z  W and x  C, we have f (x, z)  . Hence F (W)  S (C, ), as desired.
29.2 Summary

 Give a point x of the set X and an open set U of the space Y, let
S(x, U) = {f | f  Y and f(x)  U}
X
The sets S(x, U) are a sub-basis for topology on Y , which is called the topology of pointwise
X
convergence.
X
 Let (Y, d) be a metric space; let X be a topological space. Given an element f of Y , a compact
subspace C of X, and a number  > 0, let B (f, ) denote the set of all those elements g of Y X
C
for which
sup{d(f(x), g(x))|x  C} < 

The sets B (f, ) form a basis for a topology of Y . It is called the topology of compact
X
C
convergence.
 A space X is said to be compactly generated if it satisfies the following condition. A set A
is open in X if A C is open in C for each compact subspace C of X. This condition
is equivalent to requiring that a set B be closed in X if B C is closed in C for each
compact C. It is a fairly mild restriction on the space; many familiar spaces are compactly
generated.

 Let X and Y be topological spaces if C is a compact subspace of X and U is an open subset
of Y, define S(C, U) = {f | f  C(x, y) and f(C)  U}.

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