Constructive strong regularity and the extension property of a compactification

In contexts in which the principle of dependent choice may not be available, as toposes or Constructive Set Theory, standard locale theoretic results related to complete regularity fail hold. To resolve this difficulty, B. Banaschewski and A. Pultr introduced strongly regular locales. Unfortunately, Pultr's notion relies on non-constructive set existence principles that hinder its use Theory. article, a fully constructive formulation strong for locales is by replacing with coinductive definitions, exploiting Relation Reflection Scheme. As an application, every L proved have compact compactification. The construction compactification then used derive main result article: characterization compactifications (and thus, classically, space) terms their ability extending continuous functions codomains. Finally, open problem reflection presented.