Local (topology)

In mathematical topology , one says that a property of topological spaces applies locally to a topological space  if, for each choice of a point  in a surrounding basis of , the elements have the property. ${\displaystyle T}$ ${\displaystyle x}$ ${\displaystyle T}$ ${\displaystyle x}$

A property of topological spaces is called local if it corresponds to the associated local property.

Examples

Local properties:

• continuity

Often the local property is weaker than the original:

• locally contractible is weaker than contractible
• lokalkompakt is weaker than compact
• local Hausdorff spaces are not necessary Hausdorff spaces

Sometimes the local property is stronger than the original:

• locally singly connected is stronger than semilocal singly connected

In general, the local property is neither stronger nor weaker:

• The ridge is path-connected , but not locally connected , the discretely topologized two-element space is locally connected , but not path-connected .
• A system of subsets of a topological space is called locally finite if every point has a neighborhood that only touches a finite number of the subsets.
• A topological space can be locally metrizable if every point has a metrizable neighborhood.

Web links

• Eric W. Weisstein : Local . In: MathWorld (English).