semicontinuous

Suppose $X$ is a topological space, and $f$ is a function from $X$ 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. If is 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 with respect to the topology for ${ℝ}^{*}$ containing $\mathrm{\varnothing }$ and open 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 than the usual topology for the extended numbers. However, the sets $U(\alpha )$ can be seen as neighborhoods of infinity, so in 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 if it 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 that equals $\mathrm{\infty }$ in the set and $0$ outside.

   It follows that the characteristic function of $\mathbb{Q}$ is not semicontinuous.

3. On $ℝ$, the function $f(x)=1/x$ for $x\ne 0$ and $f(0)=0$, is not semicontinuous. 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.

Title	semicontinuous
Canonical name	Semicontinuous1
Date of creation	2013-03-22 14:00:16
Last modified on	2013-03-22 14:00:16
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	13
Author	bwebste (988)
Entry type	Definition
Classification	msc 26A15
Defines	lower semicontinuous
Defines	upper semicontinuous
Defines	lower semi-continuous
Defines	upper semi-continuous