# Difference between revisions of "Locally operator"

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* 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 <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 | + | * If all spaces satisfying the property <math>p</math> are [[self-based space]]s, then being locally <math>p</math> is equivalent 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>. | * 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>. | * 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>. | * 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>. |

## Revision as of 14:17, 6 July 2019

Template:Topospace property modifier

## Definition

### Global "locally" operator

Let be a property of topological spaces. The property *locally* is defined as follows. A topological space is termed locally if it satisfies the following equivalent conditions:

- has a basis such that all the members of the basis satisfy property when endowed with the subspace topology.
- For any point and any open subset such that , there exists an open subset of such that , and satisfies when given the subspace topology from .

### "locally" at a point

Let be a property of topological spaces. A topological space is said to satisfy *locally* at a point if the following holds: for any open subset such that , there exists an open subset of such that , and satisfies when given the subspace topology from . The key difference with the global definition is that we are now imposing the condition relative to that particular point and not for all points.

## Facts

- If is a subspace-hereditary property of topological spaces, then being locally is equivalent to the condition that every point is contained in an open subset satisfying . An example of this is the case that is the property of being a Hausdorff space, and locally is the property of being a locally Hausdorff space.
- If all spaces satisfying the property are self-based spaces, then being locally is equivalent to the condition that every point is contained an an open subset satisfying . An example of this is the case that is the property of being a Euclidean space, and locally is the property of being a locally Euclidean space.
- If is a subspace-hereditary property of topological spaces, then implies locally .
- 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 implies locally . - If is hereditary on open subsets, a regular space is locally iff it is strongly locally .