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{{r|Closed set}}

...mplement]] <math>S = R \setminus \mathfrak{p}</math> is a multiplicatively closed set and the localisation of ''R'' at <math>\mathfrak{p}</math> is the localisat

...is, if <math>\scriptstyle \overline{A}=X</math>); equivalently, the only [[closed set]] in ''X'' containing ''A'' is ''X'' itself.

{{r|Closed set}}

...be continuous if for any open set <math>U \in O_Y</math> (respectively, [[closed set|closed subset]] of ''Y'' ) the set <math>f^{-1}(U)=\{ x \in X \mid f(x) \in

{{r|Closed set}}

{{r|Closed set}}

{{r|Closed set}}

...pace|noetherian]]'' if it satisfies the [[descending chain condition]] for closed set: any descending chain of closed subsets <math>Y_0\supseteq Y_1\supseteq\ldo

...makes it possible to define the geometric operation of projection onto a [[closed set|closed]] subspace (in the metric topology induced by the inner product), ju

...n ''their'' definitions, etc. Meaning cannot emerge from definitions in a closed set of words. Without some ‘starter’ words whose meanings do not depend on

...s, and in fact one cannot define them in simpler words, without creating a closed set of words with a circularity of definitions that ultimately precludes provid ...a circular task, not surprisingly, because the dictionary comprises only a closed set of words, finite in number, that enable descriptions of the meanings of eac

...Borel spaces) are especially useful. Every Borel set (in particular, every closed set and every open set) in a Euclidean space (and more generally, in a complete