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Subbasis

المؤلف:  المرجع الالكتروني للمعلوماتيه

المصدر:  www.almerja.com

الجزء والصفحة:  ...

29-7-2021

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Subbasis

A collection of subsets of a topological space that is contained in a basis of the topology and can be completed to a basis when adding all finite intersections of the subsets.

A subbasis for the Euclidean topology of the real line is formed by all intervals  and : in fact a basis is formed by the open intervals .

A subbasis for the discrete topology of the real line is formed by all subsets of  having a given number  of elements, since every singleton set {x}" src="https://mathworld.wolfram.com/images/equations/Subbasis/Inline6.gif" style="height:15px; width:17px" /> can be obtained as the intersection of the sets {x,x+1,...,x+n-1}" src="https://mathworld.wolfram.com/images/equations/Subbasis/Inline7.gif" style="height:15px; width:135px" /> and {x-n+1,x-n+2,...,x}" src="https://mathworld.wolfram.com/images/equations/Subbasis/Inline8.gif" style="height:15px; width:157px" />.

A subbasis for the Zariski topology of the affine space  is formed by the complement sets of all irreducible affine varieties. This follows applying de Morgan's laws when considering that the open sets are the complement sets of the affine varieties, and each of these is the union of a finite number of irreducible varieties.