nLab semicontinuous map

Idea

Recall that a (say real-valued) function $f$ is continuous at a point $x$ if, roughly speaking, $f(x) \approx f(y)$ (meaning that $f(x)$ is close to $f(y)$) whenever $x \approx y$. For a lower semicontinuous map, we require only $f(x) \lesssim f(y)$ (meaning that $f(x)$ is close to or less than $f(y)$); for an upper semicontinuous map, we require only $f(x) \gtrsim f(y)$.

Definitions

Let $X$ be a topological space, let $R$ be a linearly ordered set, and let $f$ be a function from $X$ to $R$.

In nonstandard analysis, the vague idea above becomes a precise definition, so long as we use the appropriate quantifiers for $x$ and $y$.

Definition

The function $f$ is lower semicontinuous if, for each standard point? $x$ of $X$ and each hyperpoint? $y$ in the infinitesimal neighbourhood of $x$, $f(y)$ is either greater than or in the infinitesimal neighbourhood of $f(x)$. Similarly, $f$ is upper semicontinuous if, for each standard point $x$ of $X$ and each hyperpoint $y$ in the infinitesimal neighbourhood of $x$, $f(y)$ is either less than or in the infinitesimal neighbourhood of $f(x)$.

In classical analysis?, we must phrase this another way:

Definition

The function $f$ is lower semicontinuous if, for each point $x$ of $X$ and each $a \lt f(x)$ in $R$, there is some neighbourhood $U$ of $x$ such that, for each $y \in U$, $f(y) \gt a$. Similarly, $f$ is upper semicontinuous if, for each point $x$ of $X$ and each $b \gt f(x)$ in $L$, there is some neighbourhood $U$ of $x$ such that, for each $y \in U$, $f(y) \lt b$.

We can also refer to an appropriate topological structure on $R$:

Definition

The function $f$ is lower semicontinuous if it is continuous from $X$ to $R$ with the lower semicontinuous topology. Similarly, $f$ is upper semicontinuous if it is continuous from $X$ to $R$ with the upper semicontinuous topology.

Of course, this doesn't really say anything if you don't know what those topologies on $R$ are, and the easiest way to figure that out is to refer to Definition (or to Definition if you know enough nonstandard analysis to interpret it).

Examples

A function is continuous (with respect to the usual order topology on $R$) iff it is both upper and lower semicontinuous.

The characteristic function of a subset $A$ (either valued in the poset of truth values with its usual order or valued in the real numbers with $1$ for true and $0$ for false) is lower semicontinuous iff $A$ is open, and upper semicontinuous iff $A$ is closed (hence continuous iff $A$ is clopen).

Semicontinuity for multi-valued maps

We define the small and large preimages of a subset $V \subset Y$ under a multi-valued function $F\colon X \to Y$ by

$$\begin{aligned} F^{*s}(V) &\coloneqq \{ x \in X \mid F(x) \subset V \} \;\text{and} \\ F^{*l}(V) &\coloneqq \{ x \in X \mid F(x) \cap V \neq \emptyset \}. \end{aligned}$$

A multi-valued function $F\colon X \to Y$ is said to be upper semicontinuous if the small preimage of all open sets are open and is said to be lower semicontinuous if the large preimages of all open sets are open. Both properties have also a point-wise variant. The map $F$ is upper semicontinuous at $x$ for some $x \in X$ if for every open neighborhood $V$ of $F(x)$ there is a neighborhood $U$ of $x$ such that for all $x'\in U$ the set $F(x')$ is contained in $V$. Likewise, $F$ is lower semicontinuous at $x$ for some $x \in X$ if for every open neighborhood $V$ intersecting $F(x)$ (i.e. $V \cap F(x) \neq \emptyset$) there is a neighborhood $U$ of $x$ such that for all $x'\in U$ the set $F(x')$ intersects $V$.

Generalizations

We need to say something about when $R$ is a dcpo or something like that (involving the Scott topology), as well as semicontinuity in constructive mathematics involving locales. Compare also the one-sided real numbers.

Related concepts

• selection theorem

References

To read later:

• Li Yong-ming and Wang Guo-jun, Localic Katětov–Tong insertion theorem and localic Tietze extension theorem, pdf.

• Gutiérrez García and Jorge Picado, On the algebraic representation of semicontinuity, doi.