nLab
semicontinuous topology

Semicontinuous topologies

Semicontinuous topologies

Idea

The (lower or upper) semicontinuous topology is a topology on the real line (or a generalization thereof) such that a continuous function (from some topological space XX) to the real line equipped with this semicontinuous topology is the same thing as a (lower or upper) semicontinuous map from XX to the real line.

Thus one replaces discussion of semicontinuous maps with continuous maps by using a different topological structure.

Definition

Let RR be any linear order; think of the real line with its usual order. For each element aa of RR, consider the subsets

L a{xR|x>a}, L_a \coloneqq \{ x \in R \;|\; x \gt a \} ,
R a{xR|x<a}. R_a \coloneqq \{ x \in R \;|\; x \lt a \} .
Definitions

The lower semicontinuous topology on RR is generated by the base (of open sets) given by the sets L aL_a; the upper semicontinuous topology on RR is generated by the base (of open sets) given by the sets R aR_a.

(more to come)

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 June 25, 2019 at 12:07:46. See the history of this page for a list of all contributions to it.