# 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 NetsAndClosuresOfSubspaces 2013-03-22 17:18:34 2013-03-22 17:18:34 azdbacks4234 (14155) azdbacks4234 (14155) 11 azdbacks4234 (14155) Theorem msc 54A20 Net DirectedSet PartialOrder