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An irreducible component of a topological space is a maximal irreducible subset. If a subset is irreducible, its closure is also irreducible, so irreducible components are closed.
2. Apr. 2020 · 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.
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).
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 topology.
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3. Jan. 2016 · A topological space that cannot be represented as the union of two proper closed subspaces. Equivalently, an irreducible topological space can also be defined by postulating that any open subset of it is connected or that any non-empty open subset is everywhere dense.
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.
29. Juli 2014 · 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. (ii) If X is irreducible, every non-empty open subset of X is irreducible.
