nLab locally positive locale

Definition

A locale $L$ is locally positive (or locally $(-1)$-connected, overt, or open, but not to be confused with an open sublocale) if every open sublocale $O \in \mathcal{O}(L)$ is an indexed join of positive open sublocales, or equivalently, if it has a base of positive open sublocales.

Other definitions of a locally positive locale include that the unique map $!:L \to 1$ is open, and that the product projection $\pi_2:L \times L' \to L'$ is open for every locale $L'$.

See also

• locale

• positive locale

• overt space

• premetric locale

• geometric hyperdoctrine

References

• Simon Henry, Localic Metric spaces and the localic Gelfand duality (arXiv:1411.0898v1)

• Graham Manuell, Uniform locales and their constructive aspects, (arXiv:2106.00678)