nets and closures of subspaces
Let be a topological space, a point of , and a subspace of . Suppose first that , and let be the collection of neighborhoods of , http://planetmath.org/node/123partially ordered by reverse . For each , select a point (such a point is guaranteed to exist because ); then is a net of points in , and we claim that . To see this, let be a neighborhood of in , and note that, by construction, ; furthermore, if satisfies , then because , . It follows that . Conversely, suppose there exists a net of points of converging to , and let be a neighborhood of . Since , there exists such that whenever . Because for each by hypothesis, we may conclude that , hence that . ∎
The forward implication of the preceding is a generalization of the result that a point of a topological space is in the closure of a subspace if there is a sequence of points of the subspace converging to the point, as a sequence is just a net with the positive integers as its http://planetmath.org/node/360domain; however, the converse (if a point is in the closure of a subspace then there exists a sequence of points of the subspace converging to the point) requires the additional condition that the ambient topological space be first countable.
|Title||nets and closures of subspaces|
|Date of creation||2013-03-22 17:18:34|
|Last modified on||2013-03-22 17:18:34|
|Last modified by||azdbacks4234 (14155)|