separation axioms

The separation axioms are a list of (originally four, now more) properties of a topological space, all of which are satisfied by metric spaces. They all have to do with saying that two sets (of certain forms) in the space are ‘separated’ from each other in one sense if they are ‘separated’ in a (generally) weaker sense. Often, the axioms can also be interpreted in a broader context, such as in a convergence space or in a locale, or under weaker assumptions, such as those of constructive mathematics and predicative mathematics.

First, we will consider how, for topological spaces in classical mathematics, the separation axioms are about sets' being ‘separated’ as stated above. Throughout, fix a topological space $S$.

Fix two sets (subsets) $F$ and $G$ of $S$.

- The sets $F$ and $G$ are
**disjoint**if their intersection is empty:$F \cap G = \empty .$ - They are
**topologically disjoint**if there exists a neighbourhood of one set that is disjoint from the other set:$(\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \empty) \;\vee\; (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \empty) .$Notice that topologically disjoint sets must be disjoint.

- They are
**separated**if each set has a neighbourhood that is disjoint from the other set:$(\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \empty) \;\wedge\; (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \empty) \;\;\equiv\;\; \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap G = \empty \;\wedge\; F \cap V = \empty .$Notice that separated sets must be topologically disjoint.

- They are
**separated by neighbourhoods**if they have disjoint neighbourhoods:$\exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap V = \empty .$Notice that sets separated by neighbourhoods must be separated.

- They are
**separated by closed neighbourhoods**if they have disjoint closed neighbourhoods:$\exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; Cl(U) \cap Cl(V) = \empty .$Notice that sets separated by closed neighbourhoods must be separated by neighbourhoods.

- They are
**separated by a function**if there exists a continuous real-valued function on the space that maps $F$ to $0$ and $G$ to $1$:$\exists\; f: S \to \mathbf{R},\; F \subseteq f^*(\{0\}) \;\wedge\; G \subseteq f^*(\{1\}) .$Notice that sets separated by a function must be separated by closed neighbourhoods (the preimages of $[-\epsilon, \epsilon]$ and $[1-\epsilon, 1+\epsilon]$).

- Finally, they are
**precisely separated by a function**if there exists a continuous real-valued function on the space that maps precisely $F$ to $0$ and $G$ to $1$:$\exists\; f: S \to \mathbf{R},\; F = f^*(\{0\}) \;\wedge\; G = f^*(\{1\}) .$Notice that sets precisely separated by a function must be separated by a function.

Often $F$ and $G$ will be points (identified with their singleton subsets); in that case, one usually says *distinct* in place of *disjoint*.

Often $F$ or $G$ will be closed sets; notice that disjoint closed sets are automatically separated, while a closed set and a point, if disjoint, are automatically topologically disjoint.

The classical separation axioms are all statements of the form

- When $F$ is a (point/closed) set and $G$ is a (point/closed) set, if $F$ and $G$ are (separated in some weak sense), then they are (separated in some strong sense).

The axioms with names (at least with known to the authors so far of this article) are summarised in the tables below. When a row or column is missing from a table, either no name is known or the implication follows from the converses mentioned after the separation conditions above in the context of that table; there are two potential tables that are completely blank for the latter reason. When an entry in a table is repeated, that corresponds to a theorem that one separation axiom implies another.

When both sets are points:

Stronger condition ↓\Weaker condition → | Distinct | Topologically distinct |
---|---|---|

Topologically distinct | $T_0$ | |

Separated | $T_1$ | $R_0$ |

Separated by neighbourhoods | $T_2$ | $R_1$ |

Separated by closed neighbourhoods | $T_{2\frac{1}{2}}$ | $R_{1\frac{1}{2}}$ |

Separated by a function | Completely $T_2$ | Completely $R_1$ |

When one set is a point and the other is closed:

Stronger condition ↓\Weaker condition → | Disjoint |
---|---|

Separated by neighbourhoods | Regular |

Separated by closed neighbourhoods | Regular |

Separated by a function | Completely regular |

When both sets are closed:

Stronger condition ↓\Weaker condition → | Disjoint |
---|---|

Separated by neighbourhoods | Normal |

Separated by closed neighbourhoods | Normal |

Separated by a function | Normal |

Precisely separated by a function | Perfectly normal |

When the sets are arbitrary:

Stronger condition ↓\Weaker condition → | Separated |
---|---|

Separated by neighbourhoods | Completely normal |

First of all, notice that the $T_1$ condition, that distinct points are separated, is equivalent to the condition that every point is closed. Thus, $T_1$ serves as a linchpin between conditions on points and conditions on closed sets.

Many implications between separation axioms can be seen in the following Hasse diagram:

Here, there are two entries at each node; the one on the right includes the $T_0$ axiom, while the one on the left does not. This diagram shows the separation axioms as a meet sub-semilattice of the lattice of all conditions on topological spaces; for example, you can see, by following the diagram upwards, that any space that is both **n**ormal and **r**egular must be $R_3$. And since $R_3$ never appears in the tables above, you can take this as a *definition* of $R_3$.

In general, the names in this diagram are:

- ‘P’ for ‘perfectly’,
- ‘C’ for ‘complete’,
- ‘N’ for ‘normal’,
- ‘R’ (without a subscript) for ‘regular’
- ‘T’ or ‘R’ with a subscript are written and pronounced that way.

Warning: $T_i$ for $i \geq 3$ has been used in different ways in the past, and perhaps by some schools still. Also, all of the $R_i$ terms are rare. It is safest to say, for example, ‘normal Hausdorff’ for $T_4$ and clearer to say, for example, ‘normal regular’ for $R_3$. If you want to avoid the subscript terms entirely, then you can, by doing the above and the following:

- $T_2$ = Hausdorff,
- $T_1$ = accessible,
- $T_0$ = Kolmogorov,
- $R_1$ = reciprocal,
- $R_0$ = symmetric.

On the other hand, if you want to use *more* symbols, then you can:

- $R_2$ = regular,
- $R_{2\frac{1}{2}}$ = completely regular.

It would be easy to invent an $N_i$ series for the various kinds of normal spaces, but nobody seems to have done so yet.

Other terms are also in use, principally ‘Tychonoff’ for completely regular Hausdorff ($T_{3\frac{1}{2}}$).

There are other axioms sometimes included among the separation axioms that don't fit the preceding pattern; but like the others, they all hold of a metric space:

- sober and having enough points,
- semiregular,
- fully normal and fully $T_4$, which are related to paracompactness.

The axioms $T_1$ and below can be phrased entirely in terms of the specialisation order, as follows:

- In general, the specialisation order is a preorder.
- The space is $T_0$ if and only if the specialisation order is a partial order.
- The space is $R_0$ if and only if the specialisation order is an equivalence relation.
- The space is $T_1$ if and only if the sepcification order is the equality relation.

Note that *any* preorder is the specialisation order for its own specialisation topology.

The separation conditions that appear in $T_2$ and below, or rather their negations, can be easily phrased in terms of the convergence structure, as follows:

- Two points are not distinct if and only if they are equal (of course).
- They are
**topologically indistinguishable**(that is, not topologically distinct) if and only if every net (or filter) that converges to one must also converge to the other; it's enough to check the ultrafilters generated by the two points. - They are not separated if and only there exists a net (or proper filter) that converges to both.

So by taking contrapositives, it's easy to generalise $T_2$ and below to convergence spaces. (All of the axioms *can* be generalised to convergence spaces, since the convergence structure determines the topology, but there are several ways to do so, and it's not clear in general which is best.)

For locales, the axioms at the other end are clearest. Here we want to put everything in terms of open sets, so we simply work with the complements of the closed sets that appear in those axioms. Rather than talk about a closed set $F$ and a neighbourhood $U$ of $F$, we talk about an open set $G$ and an open set $U$ such that $G \cup U$ is the entire space. Now the axioms at the low end are tricky, although there is a standard answer as far down as $T_2$. (Note that every locale is $T_0$, indeed sober.)

In constructive mathematics, while the classical definitions all make sense, they are never quite what is wanted. For the low axioms, one may use, as with convergence spaces, conditions that are classically the negations of the separation conditions; for the high axioms, one may use the open sets that are classically the complements of the closed sets in the axioms. In the middle axioms, these work together; for example, the condition that a point $x$ is disjoint from a closed set $F$ becomes the condition that $x$ belongs to an open set $G$.

Specific examples should be found on the pages for specific separation axioms.

The English Wikipedia has a decent article, but since I wrote that too, it's not really an independent source. But you can check the references there!

Revised on February 11, 2016 18:32:40
by David Roberts
(202.138.13.130)