# Separation axioms

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In topology, **separation axioms** describe classes of topological space according to how well the open sets of the topology distinguish between distinct points.

## Terminology

A *neighbourhood of a point* *x* in a topological space *X* is a set *N* such that *x* is in the interior of *N*; that is, there is an open set *U* such that .
A *neighbourhood of a set* *A* in *X* is a set *N* such that *A* is contained in the interior of *N*; that is, there is an open set *U* such that .

Subsets *U* and *V* are *separated* in *X* if *U* is disjoint from the closure of *V* and *V* is disjoint from the closure of *U*.

A **Urysohn function** for subsets *A* and *B* of *X* is a continuous function *f* from *X* to the real unit interval such that *f* is 0 on *A* and 1 on *B*.

## Axioms

A topological space *X* is

**T0**if for any two distinct points there is an open set which contains just one**T1**if for any two points*x*,*y*there are open sets*U*and*V*such that*U*contains*x*but not*y*, and*V*contains*y*but not*x***T2**if any two distinct points have disjoint neighbourhoods**T2½**if distinct points have disjoint closed neighbourhoods**T3**if a closed set*A*and a point*x*not in*A*have disjoint neighbourhoods**T3½**if for any closed set*A*and point*x*not in*A*there is a Urysohn function for*A*and {*x*}**T4**if disjoint closed sets have disjoint neighbourhoods**T5**if separated sets have disjoint neighbourhoods

**Hausdorff**is a synonym for T2**completely Hausdorff**is a synonym for T2½

**regular**if T0 and T3**completely regular**if T0 and T3½**Tychonoff**is completely regular and T1

**normal**if T0 and T4**completely normal**if T1 and T5**perfectly normal**if normal and every closed set is a G_{δ}

## Properties

- A space is T1 if and only if each point (singleton) forms a closed set.
*Urysohn's Lemma*: if*A*and*B*are disjoint closed subsets of a T4 space*X*, there is a Urysohn function for*A*and*B'*.

## References

- Steen, Lynn Arthur & J. Arthur Jr. Seebach (1978),
*Counterexamples in Topology*, Berlin, New York: Springer-Verlag, ISBN 0-387-90312-7