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 .
Fix two sets (subsets) and of .
Notice that topologically disjoint sets must be disjoint.
Notice that separated sets must be topologically disjoint.
Notice that sets separated by neighbourhoods must be separated.
Notice that sets separated by closed neighbourhoods must be separated by neighbourhoods.
Notice that sets separated by a function must be separated by closed neighbourhoods (the preimages of and ).
Notice that sets precisely separated by a function must be separated by a function.
Often and will be points (identified with their singleton subsets); in that case, one usually says distinct in place of disjoint.
Often or 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
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|
|Separated by neighbourhoods|
|Separated by closed neighbourhoods|
|Separated by a function||Completely||Completely|
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 condition, that distinct points are separated, is equivalent to the condition that every point is closed. Thus, 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 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 normal and regular must be . And since never appears in the tables above, you can take this as a definition of .
In general, the names in this diagram are:
Warning: for has been used in different ways in the past, and perhaps by some schools still. Also, all of the terms are rare. It is safest to say, for example, ‘normal Hausdorff’ for and clearer to say, for example, ‘normal regular’ for . If you want to avoid the subscript terms entirely, then you can, by doing the above and the following:
On the other hand, if you want to use more symbols, then you can:
It would be easy to invent an 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 ().
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:
The axioms and below can be phrased entirely in terms of the specialisation order, as follows:
Note that any preorder is the specialisation order for its own specialisation topology.
So by taking contrapositives, it's easy to generalise 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 and a neighbourhood of , we talk about an open set and an open set such that is the entire space. Now the axioms at the low end are tricky, although there is a standard answer as far down as . (Note that every locale is , 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 is disjoint from a closed set becomes the condition that belongs to an open set .
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!