# Normal space

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*Please see* Convention:Hausdorffness assumption

## Definition

### Equivalent definitions in tabular format

Note that under some conventions, the definition below is taken as the definition of *normal space*. However, under the convention followed in this wiki, the definition of normal space includes the assumption that points are closed, forcing the space to be a T1 space (and further also a Hausdorff space, based on the definition). The definition given here is obtained when we relax the requirement of points being closed. Note that this definition *includes* the normal spaces where all points are closed, but also includes some other spaces.

No. | Shorthand | A topological space is said to be normal(-minus-Hausdorff) if ... | A topological space is said to be normal(-minus-Hausdorff) if ... |
---|---|---|---|

1 | separation of disjoint closed subsets by open subsets | given any two disjoint closed subsets in the topological space, there are disjoint open sets containing them. | given any two closed subsets such that , there exist disjoint open subsets of such that , and . |

2 | separation of disjoint closed subsets by continuous functions | given any two disjoint closed subsets, there is a continuous function taking the value at one closed set and 1 at the other. | for any two closed subsets , such that , there exists a continuous map (to the closed unit interval) such that and . |

3 | point-finite open cover has shrinking | every point-finite open cover possesses a shrinking. | for any point-finite open cover of , there exists a shrinking : the form an open cover and . |