Esakia space

In mathematics, '''Esakia spaces''' are special ordered topological spaces introduced and studied by Leo Esakia in 1974.<ref>Esakia (1974)</ref> Esakia spaces play a fundamental role in the study of Heyting algebras, primarily by virtue of the Esakia duality—the dual equivalence between the category of Heyting algebras and the category of Esakia spaces.

==Definition==

For a partially ordered set {{math|(''X'', ≤)}} and for {{math|''x''∈ ''X''}}, let {{math|↓''x'' {{=}} {''y''∈ ''X'' : ''y''≤ ''x''}}} and let {{math|↑''x'' {{=}} {''y''∈ ''X'' : ''x''≤ ''y''}}}. Also, for {{math|''A''⊆ ''X''}}, let {{math|↓''A'' {{=}} {''y''∈ ''X'' : ''y'' ≤ ''x'' for some ''x''∈ ''A''}}} and {{math|↑''A'' {{=}} {''y''∈ ''X'' : ''y''≥ ''x'' for some ''x''∈ ''A''}}}.

An '''Esakia space''' is a Priestley space {{math|(''X'',''τ'',≤)}} such that for each clopen subset {{math|''C''}} of the topological space {{math|(''X'',''τ'')}}, the set {{math|↓''C''}} is also clopen.

==Equivalent definitions==

There are several equivalent ways to define Esakia spaces.

Theorem:<ref>Esakia (1974), Esakia (1985).</ref> Given that {{math|(''X'',''τ'')}} is a Stone space, the following conditions are equivalent:

:(i) {{math|(''X'',''τ'',≤)}} is an Esakia space.

:(ii) {{math|↑''x''}} is closed for each {{math|''x''∈ ''X''}} and {{math|↓''C''}} is clopen for each clopen {{math|''C''⊆ ''X''}}.

:(iii) {{math|↓''x''}} is closed for each {{math|''x''∈ ''X''}} and {{math|↑cl(''A'') {{=}} cl(↑''A'')}} for each {{math|''A''⊆ ''X''}} (where {{math|cl}} denotes the closure in {{math|''X''}}).

:(iv) {{math|↓''x''}} is closed for each {{math|''x''∈ ''X''}}, the least closed set containing an up-set is an up-set, and the least up-set containing a closed set is closed.

Since Priestley spaces can be described in terms of spectral spaces, the Esakia property can be expressed in spectral space terminology as follows: The Priestley space corresponding to a spectral space {{math|''X''}} is an Esakia space if and only if the closure of every constructible subset of {{math|''X''}} is constructible.<ref>see section 8.3 of Dickmann, Schwartz, Tressl (2019)</ref>

==Esakia morphisms==

Let {{math|(''X'',≤)}} and {{math|(''Y'',≤)}} be partially ordered sets and let {{math|f : ''X'' → ''Y''}} be an order-preserving map. The map {{math|f}} is a bounded morphism (also known as p-morphism) if for each {{math|''x''∈ ''X''}} and {{math|''y''∈ ''Y''}}, if {{math|f(''x'')≤ ''y''}}, then there exists {{math|''z''∈ ''X''}} such that {{math|''x''≤ ''z''}} and {{math|f(''z'') {{=}} ''y''}}.

Theorem:<ref>Esakia (1974), Esakia (1985).</ref> The following conditions are equivalent:

:(1) {{math|f}} is a bounded morphism.

:(2) {{math|f(↑''x'') {{=}} ↑f(''x'')}} for each {{math|''x''∈ ''X''}}.

:(3) {{math|f−1(↓''y'') {{=}} ↓f−1(''y'')}} for each {{math|''y''∈ ''Y''}}.

Let {{math|(''X'', ''τ'', ≤)}} and {{math|(''Y'', ''{{prime|τ}}'', ≤)}} be Esakia spaces and let {{math|f : ''X'' → ''Y''}} be a map. The map {{math|f}} is called an ''Esakia morphism'' if {{math|f}} is a continuous bounded morphism.

==Notes==

==References==

* Esakia, L. (1974). Topological Kripke models. ''Soviet Math. Dokl.'', 15 147–151. * Esakia, L. (1985). ''Heyting Algebras I. Duality Theory (Russian)''. Metsniereba, Tbilisi. * {{cite book | last1=Dickmann | first1=Max | last2=Schwartz | first2= Niels | last3=Tressl | first3= Marcus | title=Spectral Spaces| doi=10.1017/9781316543870 | year=2019 | publisher=Cambridge University Press | series=New Mathematical Monographs | volume=35 | location=Cambridge | isbn=9781107146723 }}

Category:General topology