nets and closures of subspaces

Theorem.

A point of a topological space is in the closure of a subspace if and only if there is a net of points of the subspace converging to the point.

Proof.

Let $X$ be a topological space, $x$ a point of $X$ , and $A$ a subspace of $X$ . Suppose first that $x\in\bar{A}$ , and let $\mathcal{U}$ be the collection of neighborhoods of $x$ , http://planetmath.org/node/123partially ordered by reverse . For each $U\in\mathcal{U}$ , select a point $x_{U}\in U\cap A$ (such a point is guaranteed to exist because $x\in\bar{A}$ ); then $(x_{U})_{U\in\mathcal{U}}$ is a net of points in $A$ , and we claim that $x_{U}\rightarrow x$ . To see this, let $V$ be a neighborhood of $x$ in $X$ , and note that, by construction, $x_{V}\in V$ ; furthermore, if $U\in\mathcal{U}$ satisfies $V\supset U$ , then because $x_{U}\in U$ , $x_{U}\in V$ . It follows that $x_{U}\rightarrow x$ . Conversely, suppose there exists a net $(x_{\alpha})_{\alpha\in J}$ of points of $A$ converging to $x$ , and let $U\subset X$ be a neighborhood of $x$ . Since $x_{\alpha}\rightarrow x$ , there exists $\beta\in J$ such that $x_{\alpha}\in U$ whenever $\beta\preceq\alpha$ . Because $x_{\alpha}\in A$ for each $\alpha\in J$ by hypothesis, we may conclude that $U\cap A\neq\emptyset$ , hence that $x\in\bar{A}$ . ∎

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
Canonical name	NetsAndClosuresOfSubspaces
Date of creation	2013-03-22 17:18:34
Last modified on	2013-03-22 17:18:34
Owner	azdbacks4234 (14155)
Last modified by	azdbacks4234 (14155)
Numerical id	11
Author	azdbacks4234 (14155)
Entry type	Theorem
Classification	msc 54A20
Related topic	Net
Related topic	DirectedSet
Related topic	PartialOrder