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)
History

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 – Space such that every point has a Hausdorff neighborhood
• Locally metrizable space – Topological space that is homeomorphic to a metric spacePages displaying short descriptions of redirect targets
• Monotonically normal space – Property of topological spaces stronger than normality
• Normal space – Type of topological space
• 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.