Locally normal space

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 space is locally regular and hence 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.

Theorems

Theorem 1

If "X" is homeomorphic to "Y" and "X" is locally normal, then so is "Y".

Proof

This follows from the fact that the image of a normal space under a homeomorphism is always normal.

ee also

*Locally Hausdorff space
*Locally compact space
*Locally metrizable space
*Normal space
*Homeomorphism
*Locally regular space

References