semicontinuous

Suppose $X$ is a topological space, and $f$ is a functionfrom $X$ into the extended real numbers ${ℝ}^{*}$; $f:X\to {ℝ}^{*}$.Then:

1. If$f^{-1}((\alpha ,\mathrm{\infty }])=\{x\in X\mid f(x)>\alpha \}$is an open set in $X$ for all $\alpha \in ℝ$,then $f$ is said to be lower semicontinuous.

2. Ifis an open set in $X$ for all $\alpha \in ℝ$,then $f$ is said to be upper semicontinuous.

In other words, $f$ is lower semicontinuous, if $f$ is continuous withrespect to the topology for ${ℝ}^{*}$ containing $\mathrm{\varnothing }$ andopen sets

$$U(\alpha )=(\alpha ,\mathrm{\infty }],\alpha \in ℝ\cup \{-\mathrm{\infty }\}.$$

It is not difficult to see that this is a topology. For example,for a union of sets $U({\alpha }_i)$ we have ${\cup }_{i}U({\alpha }_i)=U(inf{\alpha }_i)$.Obviously, this topology is much coarser thanthe usual topology for the extended numbers.However,the sets $U(\alpha )$ can be seen as neighborhoods of infinity, soin some sense, semicontinuous functions are ”continuous at infinity”(see example 3 below).

0.0.1 Examples

1. A function $f:X\to {ℝ}^{*}$ is continuous if and only ifit is lower and upper semicontinuous.

2. Let $f$ be the characteristic function of a set $\mathrm{\Omega }\subseteq X$.Then $f$ is lower (upper)semicontinuous if and only if $\mathrm{\Omega }$ is open (closed).This also holds for the function thatequals $\mathrm{\infty }$ in the set and $0$ outside.

   It follows that the characteristic function of $\mathbb{Q}$ is notsemicontinuous.

3. On $ℝ$, the function $f(x)=1/x$ for $x\ne 0$ and $f(0)=0$, is notsemicontinuous. This example illustrate how semicontinuous ”at infinity”.

0.0.2 Properties

Let $f:X\to {ℝ}^{*}$ be a function.

1. Restricting $f$ to a subspace preserves semicontinuity.

2. Suppose $f$ is upper (lower) semicontinuous, $A$ is a topological space, and$\mathrm{\Psi }:A\to X$ is a homeomorphism. Then $f\circ \mathrm{\Psi }$ is upper (lower) semicontinuous.

3. Suppose $f$ is upper (lower) semicontinuous, and$S:{ℝ}^{*}\to {ℝ}^{*}$ is a sense preserving homeomorphism.Then $S\circ f$ is upper (lower) semicontinuous.

4. $f$ is lower semicontinuous if and only if$-f$ is upper semicontinuous.