# Difference between revisions of "Locally operator"

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− | Let <math>p</math> be a property of topological spaces. A topological space <math>X</math> is said to satisfy <math>p</math> ''locally'' at a point <math>x \in X</math> if the following holds: for any open subset <math>V \subseteq X</math> such that <math>x \in | + | Let <math>p</math> be a property of topological spaces. A topological space <math>X</math> is said to satisfy <math>p</math> ''locally'' at a point <math>x \in X</math> if the following holds: for any open subset <math>V \subseteq X</math> such that <math>x \in V</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U \subseteq V</math>, and <math>U</math> satisfies <math>p</math> when given the subspace topology from <math>X</math>. 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== | ==Facts== |

## Latest revision as of 14:23, 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 .