# 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 $x\xe2\x88\x88\stackrel{\xc2\u017b}{A}$, and let $\mathrm{\u0111\x9d\x92\xb0}$ be the collection^{} of neighborhoods^{} of $x$, http://planetmath.org/node/123partially ordered by reverse . For each $U\xe2\x88\x88\mathrm{\u0111\x9d\x92\xb0}$, select a point ${x}_{U}\xe2\x88\x88U\xe2\x88\copyright A$ (such a point is guaranteed to exist because $x\xe2\x88\x88\stackrel{\xc2\u017b}{A}$); then ${({x}_{U})}_{U\xe2\x88\x88\mathrm{\u0111\x9d\x92\xb0}}$ is a net of points in $A$, and we claim that ${x}_{U}\xe2\x86\x92x$. To see this, let $V$ be a neighborhood of $x$ in $X$, and note that, by construction, ${x}_{V}\xe2\x88\x88V$; furthermore, if $U\xe2\x88\x88\mathrm{\u0111\x9d\x92\xb0}$ satisfies $V\xe2\x8a\x83U$, then because ${x}_{U}\xe2\x88\x88U$, ${x}_{U}\xe2\x88\x88V$. It follows that ${x}_{U}\xe2\x86\x92x$.
Conversely, suppose there exists a net ${({x}_{\mathrm{\xce\pm}})}_{\mathrm{\xce\pm}\xe2\x88\x88J}$ of points of $A$ converging to $x$, and let $U\xe2\x8a\x82X$ be a neighborhood of $x$. Since ${x}_{\mathrm{\xce\pm}}\xe2\x86\x92x$, there exists $\mathrm{\xce\u010c}\xe2\x88\x88J$ such that ${x}_{\mathrm{\xce\pm}}\xe2\x88\x88U$ whenever $\mathrm{\xce\u010c}\xe2\u0218\u017b\mathrm{\xce\pm}$. Because ${x}_{\mathrm{\xce\pm}}\xe2\x88\x88A$ for each $\mathrm{\xce\pm}\xe2\x88\x88J$ by hypothesis^{}, we may conclude that $U\xe2\x88\copyright A\xe2\x89\mathrm{\xe2\x88\x85}$, hence that $x\xe2\x88\x88\stackrel{\xc2\u017b}{A}$.
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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 |
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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 |