: Mathématiques: cover

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cover

In mathematics, and more particularly in set theory, a cover (or covering) of a set ${\\displaystyle X}$ is a family of subsets of ${\\displaystyle X}$ whose union is all of ${\\displaystyle X}$ . More formally, if ${\\displaystyle C=\\lbrace U_{\\alpha }:\\alpha \\in A\ brace }$ is an indexed family of subsets ${\\displaystyle U_{\\alpha }\\subset X}$ (indexed by the set ${\\displaystyle A}$ ), then ${\\displaystyle C}$ is a cover of ${\\displaystyle X}$ if ${\\displaystyle \igcup _{\\alpha \\in A}U_{\\alpha }=X}$ . Thus the collection ${\\displaystyle \\lbrace U_{\\alpha }:\\alpha \\in A\ brace }$ is a cover of ${\\displaystyle X}$ if each element of ${\\displaystyle X}$ belongs to at least one of the subsets ${\\displaystyle U_{\\alpha }}$ .
A subcover of a cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Cover_(topology))

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