Editing Interior algebra (section)

=== Continuous morphisms ===
The earliest generalization of continuity to interior algebras was [[Roman Sikorski|Sikorski]]'s, based on the [[inverse image]] map of a [[continuous map]]. This is a Boolean homomorphism, preserves unions of sequences and includes the closure of an inverse image in the inverse image of the closure. Sikorski thus defined a ''continuous homomorphism'' as a Boolean σ-homomorphism ''f'' between two σ-complete interior algebras such that ''f''(''x'')<sup>C</sup> ≤ ''f''(''x''<sup>C</sup>). This definition had several difficulties: The construction acts [[Functor#Covariance and contravariance|contravariantly]] producing a dual of a continuous map rather than a generalization. On the one hand σ-completeness is too weak to characterize inverse image maps (completeness is required), on the other hand it is too restrictive for a generalization. (Sikorski remarked on using non-σ-complete homomorphisms but included σ-completeness in his axioms for ''closure algebras''.) Later J. Schmid defined a '''continuous homomorphism''' or '''continuous morphism''' for interior algebras as a Boolean homomorphism ''f'' between two interior algebras satisfying ''f''(''x''<sup>C</sup>) ≤ ''f''(''x'')<sup>C</sup>. This generalizes the forward image map of a continuous map—the image of a closure is contained in the closure of the image. This construction is [[Functor#Covariance and contravariance|covariant]] but not suitable for [[category theoretic]] applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections. (C. Naturman returned to Sikorski's approach while dropping σ-completeness to produce topomorphisms as defined above. In this terminology, Sikorski's original "continuous homomorphisms" are σ-complete topomorphisms between σ-complete interior algebras.)