# Normal space

(Redirected from Normality)

## Definition

### Equivalent definitions in tabular format

No. Shorthand A topological space is said to be normal(-minus-Hausdorff) if ... A topological space $X$ 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 $A,B \subseteq X$ such that $A \cap B = \varnothing$, there exist disjoint open subsets $U,V$ of $X$ such that $A \subseteq U, B \subseteq V$, and $U \cap V = \varnothing$.
2 separation of disjoint closed subsets by continuous functions given any two disjoint closed subsets, there is a continuous function taking the value $0$ at one closed set and 1 at the other. for any two closed subsets $A,B \subseteq X$, such that $A \cap B = \varnothing$, there exists a continuous map $f:X \to [0,1]$ (to the closed unit interval) such that $f(x) = 0 \ \forall x \in A$ and $f(x) = 1 \ \forall \ x \in B$.
3 point-finite open cover has shrinking every point-finite open cover possesses a shrinking. for any point-finite open cover $U_i, i \in I$ of $X$, there exists a shrinking $V_i, i \in I$: the $V_i$ form an open cover and $\overline{V_i} \subseteq U_i$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal Hausdorff space a normal space that is also Hausdorff. Some people include Hausdorffness as part of the definition of normal space. (obvious) a set of size more than one with the trivial topology is normal but not Hausdorff |FULL LIST, MORE INFO
compact Hausdorff space a compact space that is also Hausdorff compact Hausdorff implies normal normal not implies compact Binormal space, Paracompact Hausdorff space|FULL LIST, MORE INFO
hereditarily normal space every subspace is a normal space under the subspace topology normality is not hereditary |FULL LIST, MORE INFO
paracompact Hausdorff space a paracompact space that is also Hausdorff paracompact Hausdorff implies normal normal not implies paracompact Binormal space|FULL LIST, MORE INFO
regular Lindelof space both a regular space and a Lindelof space regular Lindelof implies normal normal not implies Lindelof |FULL LIST, MORE INFO
perfectly normal space every closed subset is a G-delta subset perfectly normal implies normal normal not implies perfectly normal Hereditarily normal space|FULL LIST, MORE INFO
metrizable space can be given the structure of a metric space with the same topology metrizable implies normal normal not implies metrizable Collectionwise normal space, Elastic space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space, Paracompact Hausdorff space, Perfectly normal space, Protometrizable space|FULL LIST, MORE INFO
CW-space the underlying topological space of a CW-complex CW implies normal normal not implies CW Hereditarily normal space, Paracompact Hausdorff space, Perfectly normal space|FULL LIST, MORE INFO
linearly orderable space obtained using the order topology for some linear ordering linearly orderable implies normal normal not implies linearly orderable Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space|FULL LIST, MORE INFO
collectionwise normal space
monotonically normal space