Locally operator: Difference between revisions

Revision as of 20:18, 30 May 2016

Definition

Let $p$ be a property of topological spaces. The property locally $p$ , is defined as the property of being a topological space in which every point has an open neighbourhood, which, in the subspace topology, satisfies $p$ .

Facts

For any property $p$ of topological spaces, $p$ implies locally $p$ .
There is a variant of the locally operator, called the strongly locally operator, and the prefix adjective locally is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally $p$ implies locally $p$ .
If $p$ is hereditary on open subsets, a regular space is locally $p$ iff it is strongly locally $p$ .

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 * For any property <math>p</math> of topological spaces, <math>p</math> implies locally <math>p</math>.
 * There is a variant of the locally operator, called the [[strongly locally operator]], and the prefix adjective ''locally'' is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. Strongly locally <math>p</math> implies locally <math>p</math>.
-* If <math>p</math> is hereditary on open subsets, a [[regular space|regular]] is locally <math>p</math> iff it is strongly locally <math>p</math>.
+* If <math>p</math> is hereditary on open subsets, a [[regular space]] is locally <math>p</math> iff it is strongly locally <math>p</math>.