Semicontinuous maps

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).

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.