Locally normal space

Separation axioms
in topological spaces
Kolmogorov classification
T0 Kolmogorov
T1 Fréchet
T2 Hausdorff
T2½Urysohn
completely T2 completely Hausdorff
T3 regular Hausdorff
T3½Tychonoff
T4 normal Hausdorff
T5 completely normal
 Hausdorff
T6 perfectly normal
 Hausdorff

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

See also

Further reading

References

  1. ↑ 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.
  2. ↑ 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.