Understanding semi-continuity of multifunctions

Let \(X\) and \(Y\) be topological spaces. One of the basic objects of study in topology are the continuous functions \(f : X \to Y\). Continuity in its pure form is a topological characterization: if \(V \subseteq Y\) is an open set, then continuity is equivalent to the openness of \(f^{-1}[V] \subseteq X\). The same concept is useful for multifunctions in addition to regular functions, though we need to clarify \(f^{-1} [V]\) in this context. First, however, we need to disaggregate continuity into two weaker properties.

Lower semi-continuity

Let \(\Gamma : X \twoheadrightarrow Y\) be a multifunction. We say that \(\Gamma\) is lower semi-continuous at \(x_0 \in X\) if, for any open \(V \subseteq Y\) with

\[\Gamma x_0 \cap V \ne \emptyset ,\]

there exists an open neighborhood \(U(x_0) \subseteq X\) such that if \(x \in U(x_0)\), then

\[\Gamma x \cap V \ne \emptyset.\]

We will say that \(\Gamma\) is lower semi-continuous on \(X\) if it is lower semi-continuous at all \(x \in X\).

Here’s a visual depiction of lower semi-continuity:

You can see above that \(\Gamma x_0\) and \(\Gamma x\) need not intersect to satisfy lower semi-continuity.

Upper semi-continuity

Now let’s consider a parallel notion of semi-continuity. We say that \(\Gamma\) is upper semi-continuous at \(x_0 \in X\) if, for any open \(V \subseteq Y\) with

\[\Gamma x_0 \subseteq V,\]

there exists an open neighborhood \(U(x_0) \subseteq X\) such that if \(x \in U(x_0)\), then

\[\Gamma x \subseteq V.\]

Equivalently, this last containment is equivalent to \(\Gamma U(x_0) \subseteq V\). We will say that \(\Gamma\) is upper semi-continuous on \(X\) if it is upper semi-continuous at all \(x \in X\).[^1]

Here’s a visual depiction of upper semi-continuity:

In the above case, \(\Gamma x_0\) and \(\Gamma x\) meet, but this is not strictly necessary for \(\Gamma\) to be upper semi-continuous.

The definition I have given here for global upper semi-continuity, according to Professor Feinberg, is controversial. In Topological Spaces, Berge requires that in addition to being upper semi-continuous at every \(x \in X\), global upper semi-continuous multifunctions need also be compactly valued; that is, \(\Gamma x\) is a compact subset of \(Y\) for every \(x \in X\). I didn’t find this obviously crucial, but if you want to have the standard continuity property that \(\Gamma K\) is compact whenever \(K\) is compact, or if more basically you want to have a continuous multifunction be equivalent to both lower and upper semi-continuity, you need to throw this in. But for applications in decision theory, global compactness guarantees are really difficult in general.

The immediate distinction between lower and upper semi-continuity is clear: with lower semi-continuity we’re interested in preserving a “nonempty intersection” property, but with upper semi-continuity we’re interested in preserving a “covering” property. Okay, great. But what are we actually getting at by defining these concepts as such?

That is, I think, a challenging question in general. Nevertheless, there is one important, initial way to clarify the distinction between lower and upper semi-continuity. Namely, we can tie them back to our prior definitions of lower and upper inverses. Recall that a function \(f : X \to Y\) is continuous if and only if \(f^{-1}[G]\) is open in \(X\) whenever \(G\) is open in \(Y\). Likewise, we have the following characterizations of semi-continuous multifunctions.

Theorem. Let \(\Gamma : X \twoheadrightarrow Y\) be a multifunction. Then,

1. \(\Gamma\) is lower semi-continuous if and only if \(\Gamma^- G\) is open in \(X\) whenever \(G\) is open in \(Y\),
2. \(\Gamma\) is upper semi-continuous if and only if \(\Gamma^+ G\) is open in \(X\) whenever \(G\) is open in \(Y\).

Proof. (1.) Assume first that \(\Gamma\) is lower semi-continuous, and suppose \(G \subseteq Y\) is open. Then whenever \(x_0 \in \Gamma^- G\) we can fix an open neighborhood \(U(x_0)\) such that if \(x \in U(x_0)\) we have \(\Gamma x \cap G \ne \emptyset\) — in other words, \(x \in \Gamma^- G\). So \(U(x_0) \subseteq \Gamma^- G\), and since \(x_0\) was arbitrary we see that \(\Gamma^- G\) is an open set. Conversely assume that \(\Gamma^- G\) is open. Then, given \(x_0 \in \Gamma^- G\), we can fix an open neighbhorhood \(U(x_0) \subseteq \Gamma^- G\). But then the definition of \(\Gamma^- G\) gives us lower semi-continuity; for, \(x \in U(x_0)\) implies \(\Gamma x \cap G \ne \emptyset\).

The proof of (2.) follows a nearly identical argument to that of (1.), replacing the lower inverse criterion for the upper inverse criterion. \(\square\)

Published on June 26th, 2018 by David Kraemer