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An irreducible component of an algebraic set is an algebraic subset that is irreducible and maximal (for set inclusion) for this property. For example, the set of solutions of the equation xy = 0 is not irreducible, and its irreducible components are the two lines of equations x = 0 and y = 0 .
Irreducible components. Definition . Let be a topological space. We say X is irreducible, if X is not empty, and whenever X = Z1 ∪Z2 with Zi closed, we have X =Z1 or X =Z2. We say Z ⊂ X is an irreducible component of X if Z is a maximal irreducible subset of X. An irreducible space is obviously connected.
A topological space is irreducible if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much significance for Hausdorff spaces. See also irreducible component, algebraic variety.
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint).
The closure of an irreducible subset of an irreducible space is irreducible. Ask Question. Asked 11 years, 3 months ago. Modified 7 years, 2 months ago. Viewed 5k times. 19. Start with an irreducible space X X. Take a subset Y Y that is irreducible. Show that the closure of Y Y is still irreducible.
29. Juli 2014 · A non-empty topological space X is irreducible if every pair of non-empty open sets in X intersect (thus X is as far as possible from being Hausdorff). Equivalent conditions: (a) X is not the union of two proper closed subsets. (b) If F i ( 1 ≤ i ≤ n) are closed subsets which cover X, then X = F i for some i .
4. Irreducible sets. (4.1) Definition. A topological space X is irreducible if X is non-empty, and if any two non-empty open subsets of X intersect. Equivalently X is irreducible if X 6= ; and X is not the union of two closed subsets different from X. A subset Y of X is irreducible if it is an irreducible topological space with the induced ...
