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separation axioms

The separation axioms are additional conditions which may be required to a topological space in order to ensure that some particular types of sets can be separated by open sets, thus avoiding certain pathological cases.

Axiom	Definition
$T_0$	given two distinct points, there is an open set containing exactly one of them;
$T_1$	given two distinct points, there is a neighborhood of each of them which does not contain the other point;
$T_2$	given two distinct points, there are two disjoint open sets each of which contains one of the points;
$T_{2\frac{1}{2}}$	given two distinct points, there are two open sets, each of which contains one of the points, whose closures are disjoint;
$T_3$	given a closed set $A$ and a point $x\notin A$ , there are two disjoint open sets $U$ and $V$ such that $x\in U$ and $A\subset V$ ;
$T_{3\frac{1}{2}}$	given a closed set $A$ and a point $x\notin A$ , there is an Urysohn function for $A$ and $\{b\}$ ;
$T_4$	given two disjoint closed sets $A$ and $B$ , there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$ ;
$T_5$	given two separated sets $A$ and $B$ , there are two disjoint open sets $U$ and $V$ such that $A\subset U$ and $B\subset V$ .

If a topological space satisfies a $T_i$ axiom, it is called a $T_i$ -space. The following table shows other common names for topological spaces with these or other additional separation properties.

Name	Separation properties
Kolmogorov space	$T_0$
Fréchet space	$T_1$
Hausdorff space	$T_2$
Completely Hausdorff space	$T_{2\frac{1}{2}}$
Regular space	$T_3$ and $T_0$
Tychonoff or completely regular space	$T_{3\frac{1}{2}}$ and $T_0$
Normal space	$T_4$ and $T_1$
Perfectly $T_4$ space	$T_4$ and every closed set is a $G_\delta$ (see here)
Perfectly normal space	$T_1$ and perfectly $T_4$
Completely normal space	$T_5$ and $T_1$

The following implications hold strictly:

$\displaystyle (T_2$ $\displaystyle \text { and } T_3)$	$\displaystyle \Rightarrow T_{2\frac{1}{2}}$
$\displaystyle (T_3$ $\displaystyle \text { and } T_4)$	$\displaystyle \Rightarrow T_{3\frac{1}{2}}$
$\displaystyle T_{3\frac{1}{2}}$	$\displaystyle \Rightarrow T_3$
$\displaystyle T_5$	$\displaystyle \Rightarrow T_4$

$\displaystyle \text {Completely normal }$	$\displaystyle \Rightarrow$ $\displaystyle \text { normal }\Rightarrow \text { completely regular }$
	$\displaystyle \Rightarrow$ $\displaystyle \text { regular }\Rightarrow T_{2\frac{1}{2}}\Rightarrow T_2\Rightarrow T_1 \Rightarrow T_0$

Remark. Some authors define $T_3$ spaces in the way we defined regular spaces, and $T_4$ spaces in the way we defined normal spaces (and vice-versa); there is no consensus on this issue.

Bibliography: Counterexamples in Topology, L. A. Steen, J. A. Seebach Jr., Dover Publications Inc. (New York)

separation axioms is owned by Koro, matte, Pedro Sanchez.

How to Cite This Entry

Koro, matte, Pedro Sanchez. "separation axioms" (version 23). PlanetMath.org. Freely available at http://planetmath.org/SeparationAxioms.html.

Classification

AMS MSC:	54D10 (General topology :: Fairly general properties :: Lower separation axioms ($T_0$--$T_3$, etc.))
	54D15 (General topology :: Fairly general properties :: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.))

Pending Errata and Addenda

None. [ View all 10 ]

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Discussion

PM separation axioms by matte on 2004-10-06 09:23:43

(from the entry):
"Some authors define $ T_3$ spaces in the way we defined
regular spaces, and $ T_4$ spaces in the way we defined
normal spaces (and vice-versa); there is no consensus on
this issue."

In topology there seem to be two conventions for the
separation axioms. The first one is to define
T_0,T_1, T_2, T_3,T_4 as in this entry. And then
define
regular = T_3 + T_0
normal = T_4+T_1
completely normal = T_5+ T_0
That is T3,T4,T5 are weaker than the corresponding
regular, normal, completely normal.
This convention conincides with the definitions
on this page. A reference following this is
Steen, Seebach Counterexamples in topology.

However, there is also another convention (e.g Kelley).
In this convention normal, regular is defined as the
weaker condition, and T_3, T_4, T_5 as the stronger
condition. For example, the definition of
normal (in this convention) coincides with the above definition
of T_4,
and T4 (in this convention) coincindes with the above definition
of normal.
In conclusion, the meanings of T_4 and normal are switched
between the two conventions.

Similarly the switchings of meanings seem to take place
in the other definitions.

The problem is that both conventions are in use at PM.
This entry on separation axioms use the first one. But,

http://planetmath.org/encyclopedia/NormalTopologicalSpace.html

uses the second one.

Thus the question is that which convention should we adopt.
I would suggest the former (T_4 weaker, normal stronger).
Some motivation for this:

- First, both are used in the litterature, although I can not
say which is more popular.
- in view of linking it is simpler to link 'normal' (which
would typically be more used in this convention) than $T_4$.
- For clarity, it makes sense to adopt only one convention.
Either T_i axioms are all weaker than the corresponding
'normal, regular, etc' axioms or vice versa. This way
there will be some internal logic in the separation axiom
names.
- Any change is easy to perform now. There is not too much
material on separation axioms at the moment.

Comments?

Matte

[ reply | up ]

Re: PM separation axioms by matte on 2004-10-06 10:37:33
Re: PM separation axioms by yark on 2004-10-06 11:44:32
Re: PM separation axioms by Koro on 2004-10-06 12:06:46
- Re: PM separation axioms by matte on 2004-10-06 13:45:00
- - Re: PM separation axioms by matte on 2004-10-08 09:31:18

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