nLab
semicontinuous map

Semicontinuous maps

Idea

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

Definitions

Let XX be a topological space, let RR be a linearly ordered set, and let ff be a function from XX to RR.

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

Definition

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

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

Definition

The function ff is lower semicontinuous if, for each point xx of XX and each a<f(x)a \lt f(x) in RR, there is some neighbourhood UU of xx such that, for each yUy \in U, f(y)>af(y) \gt a. Similarly, ff is upper semicontinuous if, for each point xx of XX and each b>f(x)b \gt f(x) in LL, there is some neighbourhood UU of xx such that, for each yUy \in U, f(y)<bf(y) \lt b.

We can also refer to an appropriate topological structure on RR:

Definition

The function ff is lower semicontinuous if it is continuous from XX to RR with the lower semicontinuous topology. Similarly, ff is upper semicontinuous if it is continuous from XX to RR with the upper semicontinuous topology.

Of course, this doesn't really say anything if you don't know what those topologies on RR are, and the easiest way to figure that out is to refer to Definition 2 (or to Definition 1 if you know enough nonstandard analysis to interpret it).

Examples

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

The characteristic function of a subset AA (either valued in the poset of truth values with its usual order or valued in the real numbers with 11 for true and 00 for false) is lower semicontinuous iff AA is open, and upper semicontinuous iff AA is closed (hence continuous iff AA is clopen).

Generalizations

We need to say something about when RR 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.

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.

Last revised on July 11, 2013 at 00:31:21. See the history of this page for a list of all contributions to it.