In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. 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. 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.
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