Given a \( \sigma \)space \( X \) and its lattice of open sets \( \mathcal{O}(X) \), we will study congruences on the \( \sigma \)frame \( \mathcal{O}(X) \) representing the subspaces of \( X \) (referred to as induced congruences). We will show that when \( X \) is a \( T_D \) \( \sigma \)space, there is a bijection between the subspaces of \( X \) and the congruences induced by them.
While in the setting of frames and locales the sublocales of a given locale \( L \) have a concrete description as subsets of \( L \) (since frame homomorphisms have right adjoints), similar descriptions do not hold in the setting of \( \sigma \)frames and \( \sigma \)locales, which presents a challenge when we try to represent subspaces of a \( \sigma \)space as congruences in \( \sigma \)frames.
Nonetheless, through the (contravariant) adjunction between the category of \( \sigma \)spaces and \( \sigma \)continuous maps and the category of \( \sigma \)frames and \( \sigma \)frame homomorphisms, we can define and describe a congruence induced by a subspace \( Y\subseteq X \). This correspondence yields a map \( \pi:\mathcal{P}(X) \to\mathcal{C}(\mathcal{O}(X)) \) from the powerset of \( X \) to the congruence lattice of \( \mathcal{O}(X) \), which is injective when \( X \) is a \( T_D \) \( \sigma \)space. And when the axiom \( T_D \) is satisfied, we have \( \pi(\mathcal P(X))\subseteq\mathcal C_b(\mathcal{O}(X)) \), where \( \mathcal C_b(\mathcal{O}(X)) \) is the subset of \( \mathcal C(\mathcal{O}(X)) \) consisting of meets of complemented congruences.
