In mathematics, a topological space X with topology Ω is said to be spectral if Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...

• 1) X is compact and T₀;
• 2) The set C(X) of all compact-open subsets of (X,Ω) is a sublattice of Ω and a base for the topology. Note that "compact-open" does not mean a set that is both compact and open: such a set would be unusual in most interesting spaces. Here, it means locally compact. That is, a compact-open set is an open set whose closure is compact, where the closure of a set A is the intersection of all the closed sets that contain A;
• 3) X is sober, that is any nonempty closed set F which is not a closure of a singleton {x} is a union of two closed sets which differ from F.

Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class of well-behaved topological spaces. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... Wiktionary:Open - definition Open set (mathematics) Open (sport) - A type of competition in tennis and golf (among others) where entry is open to all qualifiers regardless of age. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... Wiktionary:Open - definition Open set (mathematics) Open (sport) - A type of competition in tennis and golf (among others) where entry is open to all qualifiers regardless of age. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, particularly in topology, a topological space X is sober if every irreducible closed subset of X is the closure of exactly one singleton of X. An irreducible closed subset of X is defined to be a nonempty closed subset of X which is not the union of two... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

External link

• see 8.3 - Definition 6 and bibliography