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separation axioms

The separation axioms

Idea

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.

The classical theory

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 SS.

Separation conditions

Fix two sets (subsets) FF and GG of SS.

  • The sets FF and GG are disjoint if their intersection is empty:
    FG=. F \cap G = \empty .
  • They are topologically disjoint if there exists a neighbourhood of one set that is disjoint from the other set:
    (UF,UG=)(VG,FV=). (\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:
    (UF,UG=)(VG,FV=)UF,VG,UG=FV=. (\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:
    UF,VG,UV=. \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:
    UF,VG,Cl(U)Cl(V)=. \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 FF to 00 and GG to 11:
    f:SR,Ff *({0})Gf *({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 [ϵ,ϵ][-\epsilon, \epsilon] and [1ϵ,1+ϵ][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 FF to 00 and GG to 11:
    f:SR,F=f *({0})G=f *({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 FF and GG will be points (identified with their singleton subsets); in that case, one usually says distinct in place of disjoint.

Often FF or GG 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.

Separation axioms

The classical separation axioms are all statements of the form

  • When FF is a (point/closed) set and GG is a (point/closed) set, if FF and GG 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 T_0
Separated T 1 T_1 R 0 R_0
Separated by neighbourhoods T 2 T_2 R 1 R_1
Separated by closed neighbourhoods T 212 T_{2\frac{1}{2}} R 112 R_{1\frac{1}{2}}
Separated by a function Completely T 2 T_2 Completely R 1 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

Relations between the axioms

First of all, notice that the T 1T_1 condition, that distinct points are separated, is equivalent to the condition that every point is closed. Thus, T 1T_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 0T_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 normal and regular must be R 3R_3. And since R 3R_3 never appears in the tables above, you can take this as a definition of R 3R_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 iT_i for i3i \geq 3 has been used in different ways in the past, and perhaps by some schools still. Also, all of the R iR_i terms are rare. It is safest to say, for example, ‘normal Hausdorff’ for T 4T_4 and clearer to say, for example, ‘normal regular’ for R 3R_3. If you want to avoid the subscript terms entirely, then you can, by doing the above and the following:

  • T 2T_2 = Hausdorff,
  • T 1T_1 = accessible,
  • T 0T_0 = Kolmogorov,
  • R 1R_1 = reciprocal,
  • R 0R_0 = symmetric.

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

  • R 2R_2 = regular,
  • R 212R_{2\frac{1}{2}} = completely regular.

It would be easy to invent an N iN_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 312T_{3\frac{1}{2}}).

Other axioms

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 4T_4, which are related to paracompactness.

Beyond the classical theory

The axioms T 1T_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 0T_0 if and only if the specialisation order is a partial order.
  • The space is R 0R_0 if and only if the specialisation order is an equivalence relation.
  • The space is T 1T_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 2T_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 2T_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 FF and a neighbourhood UU of FF, we talk about an open set GG and an open set UU such that GUG \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 2T_2. (Note that every locale is T 0T_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 xx is disjoint from a closed set FF becomes the condition that xx belongs to an open set GG.

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

References

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 June 28, 2014 17:35:33 by Toby Bartels (98.16.161.189)