Editing Interior algebra (section)

=== Topology ===
Given a [[topological space]] '''''X''''' = ⟨''X'', ''T''⟩ one can form the [[power set]] Boolean algebra of ''X'':

:{{math|1=⟨''P''(''X''), ∩, ∪, ′, ø, ''X''⟩}}

and extend it to an interior algebra

:{{math|1='''''A'''''('''''X''''') = ⟨''P''(''X''), ∩, ∪, ′, ø, ''X'', <sup>I</sup>⟩}},

where <sup>I</sup> is the usual topological interior operator. For all ''S'' ⊆ ''X'' it is defined by

:{{math|1=''S''<sup>I</sup> = ∪ {{mset|''O'' | ''O'' ⊆ ''S'' and ''O'' is open in '''''X'''''}}}}

For all ''S'' ⊆ ''X'' the corresponding closure operator is given by

:{{math|1=''S''<sup>C</sup> = ∩ {{mset|''C'' | ''S'' ⊆ ''C'' and ''C'' is closed in '''''X'''''}}}}

''S''<sup>I</sup> is the largest open subset of ''S'' and ''S''<sup>C</sup> is the smallest closed superset of ''S'' in '''''X'''''. The open, closed, regular open, regular closed and clopen elements of the interior algebra '''''A'''''('''''X''''') are just the open, closed, regular open, regular closed and clopen subsets of '''''X''''' respectively in the usual topological sense.

Every [[Completeness (order theory)|complete]] [[Atomic (order theory)|atomic]] interior algebra is [[isomorphism|isomorphic]] to an interior algebra of the form '''''A'''''('''''X''''') for some [[topological space]] '''''X'''''.  Moreover, every interior algebra can be [[embedding|embedded]] in such an interior algebra giving a representation of an interior algebra as a '''[[Field_of_sets#Topological_fields_of_sets|topological field of sets]]'''. The properties of the structure '''''A'''''('''''X''''') are the very motivation for the definition of interior algebras. Because of this intimate connection with topology, interior algebras have also been called '''topo-Boolean algebras''' or '''topological Boolean algebras'''.

Given a [[continuous map]] between two topological spaces

:{{math|1=''f''&nbsp;:&nbsp;'''''X'''''&nbsp;→&nbsp;'''''Y'''''}}

we can define a [[completeness (order theory)|complete]] topomorphism

:{{math|1='''''A'''''(''f'')&nbsp;:&nbsp;'''''A'''''('''''Y''''')&nbsp;→&nbsp;'''''A'''''('''''X''''')}}

by

:'''''A'''''(''f'')(''S'') = ''f''<sup>−1</sup>[''S'']

for all subsets ''S'' of '''''Y'''''. Every complete topomorphism between two complete atomic interior algebras can be derived in this way. If '''Top''' is the [[category of topological spaces]] and continuous maps and '''Cit''' is the [[category theory|category]] of complete atomic interior algebras and complete topomorphisms then '''Top''' and '''Cit''' are [[Dual (category theory)|dually isomorphic]] and {{math|1='''''A'''''&nbsp;:&nbsp;'''Top'''&nbsp;→&nbsp;'''Cit'''}} is a [[functor|contravariant functor]] that is a dual isomorphism of categories. '''''A'''''(''f'') is a homomorphism if and only if ''f'' is a continuous [[open map]].

Under this dual isomorphism of categories many natural topological properties correspond to algebraic properties, in particular connectedness properties correspond to irreducibility properties:

*'''''X''''' is [[empty set|empty]] if and only if '''''A'''''('''''X''''') is trivial
*'''''X''''' is [[indiscrete space|indiscrete]] if and only if '''''A'''''('''''X''''') is [[simple algebra|simple]]
*'''''X''''' is [[discrete space|discrete]] if and only if '''''A'''''('''''X''''') is Boolean
*'''''X''''' is [[almost discrete space|almost discrete]] if and only if '''''A'''''('''''X''''') is [[semisimple algebraic group|semisimple]]
*'''''X''''' is [[Alexandrov topology|finitely generated]] (Alexandrov) if and only if '''''A'''''('''''X''''') is '''operator complete''' i.e. its interior and closure operators distribute over arbitrary meets and joins respectively
*'''''X''''' is [[connected space|connected]] if and only if '''''A'''''('''''X''''') is [[directly indecomposable]]
*'''''X''''' is [[ultraconnected space|ultraconnected]] if and only if '''''A'''''('''''X''''') is [[finitely subdirectly irreducible]]
*'''''X''''' is [[compact space|compact]] ultra-connected if and only if '''''A'''''('''''X''''') is [[subdirectly irreducible]]

==== 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.

==== Neighbourhood functions and neighbourhood lattices ====
The topological concept of [[Neighbourhood (mathematics)|neighbourhood]]s can be generalized to interior algebras: An element ''y'' of an interior algebra is said to be a '''neighbourhood''' of an element ''x'' if {{math|1=''x'' ≤ ''y''<sup>I</sup>}}. The set of neighbourhoods of ''x'' is denoted by ''N''(''x'') and forms a [[Filter (mathematics)|filter]]. This leads to another formulation of interior algebras:

A '''neighbourhood function''' on a Boolean algebra is a mapping ''N'' from its underlying set ''B'' to its set of filters, such that:

#For all {{math|1=''x'' ∈ ''B'', max{{mset|''y'' ∈ ''B'' | ''x'' ∈ ''N''(''y'')}}}} exists
#For all {{math|1=''x'',''y'' ∈ ''B'', ''x'' ∈ ''N''(''y'')}} if and only if there is a {{math|1=''z'' ∈ ''B''}} such that {{math|1=''y'' ≤ ''z'' ≤ ''x''}} and {{math|1=''z'' ∈ ''N''(''z'')}}.

The mapping ''N'' of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra. Moreover, given a neighbourhood function ''N'' on a Boolean algebra with underlying set ''B'', we can define an interior operator by {{math|1=''x''<sup>I</sup> = max{{mset|y ∈ ''B'' | ''x'' ∈ ''N''(''y'')}}}} thereby obtaining an interior algebra. {{tmath|1=N(x)}} will then be precisely the filter of neighbourhoods of ''x'' in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions.

In terms of neighbourhood functions, the open elements are precisely those elements ''x'' such that {{math|1=''x'' ∈ ''N''(''x'')}}. In terms of open elements {{math|1=''x'' ∈ ''N''(''y'')}} if and only if there is an open element ''z'' such that {{math|1=''y'' ≤ ''z'' ≤ ''x''}}.

Neighbourhood functions may be defined more generally on [[semilattice|(meet)-semilattice]]s producing the structures known as [[neighbourhood lattice|neighbourhood (semi)lattice]]s. Interior algebras may thus be viewed as precisely the '''Boolean neighbourhood lattices''' i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.