Locally normal space
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.
| 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) |
Formal definition
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
- 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
- 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
- Locally metrizable space – Topological space that is homeomorphic to a metric space
- Monotonically normal space – Property of topological spaces stronger than normality
- Normal space – topological space in which every pair of disjoint closed sets has disjoint open neighborhoods
- Paranormal space
Further reading
Čech, Eduard (1937). "On Bicompact Spaces". Annals of Mathematics. 38 (4): 823–844. doi:10.2307/1968839. ISSN 0003-486X. JSTOR 1968839.
References
- Bella, A.; Carlson, N. (2018-01-02). "On cardinality bounds involving the weak Lindelöf degree". Quaestiones Mathematicae. 41 (1): 99–113. doi:10.2989/16073606.2017.1373157. ISSN 1607-3606. S2CID 119732758.
- Hansell, R. W.; Jayne, J. E.; Rogers, C. A. (June 1985). "Separation of K –analytic sets". Mathematika. 32 (1): 147–190. doi:10.1112/S0025579300010962. ISSN 0025-5793.