# Locally normal space

In

mathematics , particularlytopology , atopological space "X" is**locally normal**if intuitively it looks locally like anormal 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 thesubspace 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 thesubspace 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 compactHausdorff space is always locally normal.

* A normal space is always locally normal

* AT1 space need not be locally normal as the set of all real numbers endowed with thecofinite 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**

*Wikimedia Foundation.
2010.*

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