Locally normal space
| Separation axioms in topological spaces | |
|---|---|
| Kolmogorov classification | |
| T0 | (Kolmogorov) |
| T1 | (Fréchet) |
| T2 | (Hausdorff) |
| T2½ | (Urysohn) |
| completely T2 | (completely Hausdorff) |
| T3 | (regular Hausdorff) |
| T3½ | (Tychonoff) |
| T4 | (normal Hausdorff) |
| T5 | (completely normal Hausdorff) |
| T6 | (perfectly normal Hausdorff) |
In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space.[1] More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology.
Formal definition
[edit | edit source]A topological space X is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology.[2]
Note that not every neighbourhood of x has to be normal, but at least one neighbourhood of x has to be normal (under the subspace topology).
Note however, that if a space were called locally normal if and only if each point of the space belonged to a subset of the space that was normal under the subspace topology, then every topological space would be locally normal. This is because, the singleton {x} is vacuously normal and contains x. Therefore, the definition is more restrictive.
Examples and properties
[edit | edit source]- Every locally normal T1 space is locally regular and locally Hausdorff.
- A locally compact Hausdorff space is always locally normal.
- A normal space is always locally normal.
- A T1 space need not be locally normal as the set of all real numbers endowed with the cofinite topology shows.
See also
[edit | edit source]- Collectionwise normal space – Property of topological spaces stronger than normality
- Homeomorphism – Mapping which preserves all topological properties of a given space
- Locally compact space – Type of topological space in mathematics
- Locally Hausdorff space – Space such that every point has a Hausdorff neighborhood
- Locally metrizable space – Topological space that is homeomorphic to a metric space
- Monotonically normal space – Property of topological spaces stronger than normality
- Normal space – Type of topological space
- Lua error in Module:GetShortDescription at line 33: attempt to index field 'wikibase' (a nil value).
Further reading
[edit | edit source]Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
