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Locally operator

No change in size, 14:17, 6 July 2019
Facts
* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then being locally <math>p</math> is equivalent to the condition that every point is contained in an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Hausdorff space]], and locally <math>p</math> is the property of being a [[locally Hausdorff space]].
* If all spaces satisfying the property <math>p</math> are [[self-based space]]s, then being locally <math>p</math> is equivalent tot he to the condition that every point is contained an an open subset satisfying <math>p</math>. An example of this is the case that <math>p</math> is the property of being a [[Euclidean space]], and locally <math>p</math> is the property of being a [[locally Euclidean space]].
* If <math>p</math> is a [[subspace-hereditary property of topological spaces]], then <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]] is locally <math>p</math> iff it is strongly locally <math>p</math>.
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