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.

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.

Last revised on June 25, 2019 at 12:07:46.
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