nLab semicontinuous topology

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 $X$) to the real line equipped with this semicontinuous topology is the same thing as a (lower or upper) semicontinuous map from $X$ to the real line.

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

Definition

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

$$L_a \coloneqq \{ x \in R \;|\; x \gt a \} ,$$

$$R_a \coloneqq \{ x \in R \;|\; x \lt a \} .$$

Definitions

The lower semicontinuous topology on $R$ is generated by the base (of open sets) given by the sets $L_a$; the upper semicontinuous topology on $R$ is generated by the base (of open sets) given by the sets $R_a$.

(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.