Editing Spectral space (section)

==Properties==

Let ''X'' be a spectral space and let ''K''<sup><math>\circ</math></sup>(''X'') be as before. Then:
*''K''<sup><math>\circ</math></sup>(''X'') is a [[Lattice (order)|bounded sublattice]] of subsets of ''X''.
*Every closed [[Subspace topology|subspace]] of ''X'' is spectral.
*An arbitrary intersection of compact and open subsets of ''X'' (hence of elements from ''K''<sup><math>\circ</math></sup>(''X'')) is again spectral.
*''X'' is [[Kolmogorov space|T<sub>0</sub>]] by definition, but in general not [[T1 space|T<sub>1</sub>]].<ref>[[Alexander Arhangelskii|A.V. Arkhangel'skii]], [[L.S. Pontryagin]] (Eds.) ''General Topology I'' (1990) Springer-Verlag {{isbn|3-540-18178-4}} ''(See example 21, section 2.6.)''</ref> In fact a spectral space is T<sub>1</sub> if and only if it is [[Hausdorff space|Hausdorff]] (or T<sub>2</sub>) if and only if it is a [[boolean space]] if and only if ''K''<sup><math>\circ</math></sup>(''X'') is a [[boolean algebra]].
*''X'' can be seen as a [[pairwise Stone space]].<ref>G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, A. Kurz, (2010). "Bitopological duality for distributive lattices and Heyting algebras." ''Mathematical Structures in Computer Science'', 20.</ref>