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noetherian topological space : ウィキペディア英語版
noetherian topological space
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. It can also be shown to be equivalent that every open subset of such a space is compact, and in fact the seemingly stronger statement that ''every'' subset is compact.
== Definition ==
A topological space X is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence
: Y_1 \supseteq Y_2 \supseteq \cdots
of closed subsets Y_i of X, there is an integer m such that Y_m=Y_=\cdots.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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