Locally normal space explained

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

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 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.

Further reading

Čech. Eduard Čech. Eduard. 1937. On Bicompact Spaces. Annals of Mathematics. 38. 4. 823–844. 10.2307/1968839. 1968839 . 0003-486X.

Notes and References

1. Bella . A. . Carlson . N. . 2018-01-02 . On cardinality bounds involving the weak Lindelöf degree . Quaestiones Mathematicae . 41 . 1 . 99–113 . 10.2989/16073606.2017.1373157 . 119732758 . 1607-3606.
2. Hansell . R. W. . Jayne . J. E. . Rogers . C. A. . June 1985 . Separation of K –analytic sets . . en . 32 . 1 . 147–190 . 10.1112/S0025579300010962 . 0025-5793.