Lesson plan
Hyperconnected space
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 name irreducible space is preferred in algebraic geometry. For a topological space X the following conditions are equivalent: * No two nonempty open sets are disjoint. * X cannot be written as the union of two proper closed subsets. * Every nonempty open set is dense in X. * Every open set is connected. * The interior of every proper closed subset of X is empty. * Every subset is dense or nowhere dense in X. * No two points can be separated by disjoint neighbourhoods. A space which satisfies any one of these conditions is called hyperconnected or irreducible. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the Hausdorff property, some authors call such spaces anti-Hausdorff. The empty set is vacuously a hyperconnected or irreducible space under the definition above (because it contains no nonempty open sets). However some authors, especially those interested in applications to algebraic geometry, add an explicit condition that an irreducible space must be nonempty. An irreducible set is a subset of a topological space for which the subspace topology is irreducible.
Article summary
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 name irreducible space is preferred in algebraic geometry. For a topological space X the following conditions are equivalent: * No two nonempty open sets are disjoint. * X cannot be written as the union of two proper closed subsets. * Every nonempty open set is dense in X. * Every open set is connected. * The interior of every proper closed subset of X is empty. * Every subset is dense or nowhere dense in X. * No two points can be separated by disjoint neighbourhoods. A space which satisfies any one of these conditions is called hyperconnected or irreducible. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the Hausdorff property, some authors call such spaces anti-Hausdorff. The empty set is vacuously a hyperconnected or irreducible space under the definition above (because it contains no nonempty open sets). However some authors, especially those interested in applications to algebraic geometry, add an explicit condition that an irreducible space must be nonempty. An irreducible set is a subset of a topological space for which the subspace topology is irreducible.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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