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A topological space is called irreducible if any presentation of X as X = X1 ∪ X2 by two closed subsets implies that one of them is trivial ( X1 = X or X2 = X ). Clearly every irreducible space is connected. The converse is not always true but: Proposition: Let X be a connected topological space which has an open covering by irreducible ...
Sober space. In mathematics, a sober space is a topological space X such that every (nonempty) irreducible closed subset of X is the closure of exactly one point of X: that is, every irreducible closed subset has a unique generic point .
An irreducible space is also known as a hyperconnected space. Also see. Equivalence of Definitions of Irreducible Space; Definition:Irreducible Component; Definition:Ultraconnected Space; Results about irreducible spaces can be found here. Linguistic Note. The thinking behind applying the word irreducible to the concept of an irreducible space ...
2. Apr. 2020 · Definition 0.1. A topological space X X is called irreducible if it cannot be expressed as union of two proper closed subsets, or equivalently if any two inhabited open subsets have inhabited intersection. A subset S S of a topological space X X is an irreducible subset if S S is an irreducible topological space with the subspace topology. An ...
2. Juni 2011 · Then X is irreducible. (2) Any irreducible algebraic variety with the Zariski topology. A subset Y of a space X is irreducible if Y is irreducible in the induced topology. The following facts are not hard to prove: (i) If (F i) 1 ≤ i ≤ n is a finite closed covering of a space X, and if Y is an irreducible subset of X, then Y ⊆ F i for some i.
Definition A nonempty subset Y Y of a topological space X X is irreducible if it cannot be expressed as the union Y =Y1 ∪Y2 Y = Y 1 ∪ Y 2 of two proper subsets, each one of which is closed in Y Y. The empty set is not considered to be irreducible. I suppose this definition is made for algebraic sets and Zariski topology, but I was wondering ...
1. Mai 2014 · Generic point. A point in a topological space whose closure coincides with the whole space. A topological space having a generic point is an irreducible topological space; however, an irreducible space may have no generic point or may have many generic points. However, if the space satisfies the Kolmogorov axiom, then it can have at most one ...
