Topologies on Functions and Correspondences

Abstract

A compact valued correspondence may be regarded as a function mapping into the set of compact subsets of the range space. It is upper hemicontinuous if and only if, when regarded as a function, it is continuous with respect to one of the topologies on the space of compacta, and it is continuous (as a correspondence) if and only if, as a function, it is continuous with respect to another. If it is upper hemicontinuous and the the range space is Hausdorff, then its graph is closed, and if its graph is closed and the range space is compact, then it is upper hemicontinuous. If both the domain and range are compact and it is upper hemicontinuous, then its graph is compact. Consequently a topology on the space of upper hemicontinuous correspondences is induced by a topology on the set of compacta. With respect to a particular such topology, restriction is a continuous operation, composition is continuous when auxiliary conditions are satisfied, and the fixed point correspondence is upper hemicontinuous. The homotopy principle states that a correspondence from the cartesian product of the unit interval and a space to a second space is upper hemicontinuous if and only if it is continuous when viewed as a function from the unit interval to the set of upper hemicontinuous correspondences between the two spaces. The induced topology on the space of continuous functions is compared with standard topologies on the space of functions.

Exercises

5.1

Let X be a topological space, and let $\mathscr {F}$ be a family of functions from X to $\mathbb {R}$. The topology of uniform convergence is the topology generated by the subbase of sets $\{\, f' \in \mathscr {F}: |f'(x) - f(x)| < \varepsilon \text { for all }x \in X \,\}$ where $f \in \mathscr {F}$ and $\varepsilon > 0$. The topology of uniform convergence on compacta is the topology generated by the subbase of sets $\{\, f' \in \mathscr {F}: |f'(x) - f(x)| < \varepsilon \text { for all }x \in K \,\}$ where $f \in \mathscr {F}$, $K \subset X$ is compact, and $\varepsilon > 0$. Prove that if $\mathscr {F}= C(X, Y)$, then the topology of uniform convergence on compacta is the compact-open topology. (The notion of uniform convergence can be greatly generalized; cf. ch. 6 and 7 of Kelley (1955).)

5.2

Let X and Y be topological spaces, and let $\mathscr {F}$ be a family of functions from X to Y. The topology of pointwise convergence is the topology generated by the subbase of sets $\{\, f \in \mathscr {F}: f(x) \in V \,\}$ where $x \in X$ and $V \subset Y$ is open.

• (a) Prove that the topology of pointwise convergence is at least as coarse as the compact-open topology.

• (b) Let $f : [0,1] \rightarrow \mathbb {R}$ be a continuous function such that $f(0) = f(1) = 0$, and $f(t) \ne 0$ for some t. For $n = 1, 2, \ldots$ let $g_n$ be the function $g_n(t) = f(t^n)$. Prove that $\{g_n\}$ converges to the constant zero function pointwise, but not uniformly.

5.3

(Dini’s theorem for correspondences) For nonempty $S, S' \in \mathbb {R}$ we say that $S'$ dominates S in the strong set order , and write $S' \ge S$, if, for all $s \in S$ and $s' \in S'$, $\min \{s, s'\} \in S$ and $\max \{s, s'\} \in S'$. Let X be a topological space, and let $\{F_n\}$ be a sequence of continuous correspondences $F_n : X \rightarrow \mathbb {R}$ that is increasing, in the sense that $F_{n'}(x) \ge F_n(x)$ for all $x \in X$ and $n' \ge n$, and that converges pointwise to a continuous correspondence $F : X \rightarrow \mathbb {R}$. (Here pointwise convergence is defined by regarding $F_n$ and F as continuous functions from X to $\mathscr {H}(\mathbb {R})$.) Prove that $F_n$ converges to F in the weak upper topology.

5.4

Let X and Y be topological spaces, and let $\mathscr {F}$ be a family of continuous functions from X to Y. A sequence $\{f_n\}$ in $\mathscr {F}$ converges continuously to $f \in \mathscr {F}$ if, for all sequences $\{x_n\}$ converging to a point $x \in X$, $f_n(x_n) \rightarrow f(x)$. Prove that if $\{f_n\}$ converges to f in the compact-open topology, then it converges continuously to f.

5.5

Let X and Y be topological spaces. Let $e : C(X, Y) \times X \rightarrow Y$ be the function $e(f, x) = f(x)$. A topology on C(X, Y) is jointly continuous if e is continuous when $C(X, Y) \times X$ has the associated product topology.

• (a) Prove that if X is locally compact, then the compact-open topology is jointly continuous.

• (b) Prove that if A and B are topological spaces, $K \subset A$ and $L \subset B$ are compact, and $U \subset A \times B$ is a neighborhood of $K \times L$, then there are neighborhoods V of K and W of L such that $V \times W \subset U$.

• (c) Prove that a jointly continuous topology $\tau$ is at least as fine as the compact-open topology. (Concretely, given $f \in \mathscr {C}_{K_1,V_1} \cap \cdots \cap \mathscr {C}_{K_n, V_n}$ we need to construct a $\tau$-open U such that $f \in U \subset \mathscr {C}_{K_1,V_1} \cap \cdots \cap \mathscr {C}_{K_n, V_n}$.)