Editing Interior algebra (section)

==== Generalized topology ====

The modern formulation of topological spaces in terms of [[topological space|topologies]] of open subsets, motivates an alternative formulation of interior algebras: A '''generalized topological space''' is an [[algebraic structure]] of the form

:⟨''B'', ·, +, ′, 0, 1, ''T''⟩

where ⟨''B'', ·, +, ′, 0, 1⟩ is a Boolean algebra as usual, and ''T'' is a unary relation on ''B'' (subset of ''B'') such that:

#{{math|1=0,1 ∈ ''T''}}
#''T'' is closed under arbitrary joins (i.e. if a join of an arbitrary subset of ''T'' exists then it will be in ''T'')
#''T'' is closed under finite meets
#For every element ''b'' of ''B'', the join {{math|1=Σ{{mset|''a'' ∈''T'' | ''a'' ≤ ''b''}}}} exists

''T'' is said to be a '''generalized topology''' in the Boolean algebra.

Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological space

:⟨''B'', ·, +, ′, 0, 1, ''T''⟩

we can define an interior operator on ''B'' by {{math|1=''b''<sup>I</sup> = Σ{{mset|''a'' ∈''T'' | ''a'' ≤ ''b''}}}} thereby producing an interior algebra whose open elements are precisely ''T''. Thus generalized topological spaces are equivalent to interior algebras.

Considering interior algebras to be generalized topological spaces, topomorphisms are then the standard homomorphisms of Boolean algebras with added relations, so that standard results from [[universal algebra]] apply.