locally normal space

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.{{Cite journal |last1=Hansell |first1=R. W. |last2=Jayne |first2=J. E. |last3=Rogers |first3=C. A. |date=June 1985 |title=Separation of K –analytic sets |url=https://onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300010962 |journal=Mathematika |language=en |volume=32 |issue=1 |pages=147–190 |doi=10.1112/S0025579300010962 |issn=0025-5793|url-access=subscription }}

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.

• 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.

• {{annotated link|Collectionwise normal space}}
• {{annotated link|Homeomorphism}}
• {{annotated link|Locally compact space}}
• {{annotated link|Locally Hausdorff space}}
• {{annotated link|Locally metrizable space}}
• {{annotated link|Monotonically normal space}}
• {{annotated link|Normal space}}
• {{annotated link|Paranormal space}}

{{Cite journal|last=Čech|authorlink=Eduard Čech|first=Eduard|date=1937|title=On Bicompact Spaces|url=http://dx.doi.org/10.2307/1968839|journal=Annals of Mathematics|volume=38|issue=4|pages=823–844|doi=10.2307/1968839|jstor=1968839 |issn=0003-486X|url-access=subscription}}

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Category:Topology

Category:Properties of topological spaces

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