Irreducible not implies Noetherian: Difference between revisions
(New page: ==Statement== An irreducible space need not be Noetherian. ==Example== Consider a topological space whose underlying set is uncountable, and where the proper closed subsets are prec...) |
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{{topospace property non-implication}} | |||
==Statement== | ==Statement== | ||
An [[irreducible space]] need not be Noetherian. | An [[irreducible space]] need not be [[Noetherian space|Noetherian]]. | ||
==Example== | ==Example== | ||
Consider a topological space whose underlying set is uncountable, and where the proper closed subsets are precisely the countable subsets (the [[cocountable topology]]). The topological space is clearly irreducible, because a union of proper closed subsets is countable, and hence again proper. However, it is not Noetherian, because one can have an infinite descending chain of closed subsets. | Consider a topological space whose underlying set is uncountable, and where the proper closed subsets are precisely the countable subsets (the [[cocountable topology]]). The topological space is clearly irreducible, because a union of proper closed subsets is countable, and hence again proper. However, it is not Noetherian, because one can have an infinite descending chain of closed subsets. | ||
Revision as of 20:15, 16 January 2008
This article gives the statement and possibly, proof, of a non-implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property need not satisfy the second topological space property
View a complete list of topological space property non-implications | View a complete list of topological space property implications |Get help on looking up topological space property implications/non-implications
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Statement
An irreducible space need not be Noetherian.
Example
Consider a topological space whose underlying set is uncountable, and where the proper closed subsets are precisely the countable subsets (the cocountable topology). The topological space is clearly irreducible, because a union of proper closed subsets is countable, and hence again proper. However, it is not Noetherian, because one can have an infinite descending chain of closed subsets.