Topology: separation axioms

Introduction

I first came across the separation axioms in a functional analysis text (Bachman & Narici, “Functional Analysis”; Dover 1998, orig. 1966). I really like classification theorems, and these seemed really cool. As I said in the second post about general topology books, there is still not general agreement on the terminology. The mathematics is unambiguous, but there are two sets of intertwined terminology.

For example, the terms T4 and normal (to follow) are combined with the term T1 in either of two ways. T1 is unambiguous, but we either say that a topological space is

normal iff it is T1 and T4

or

T4 iff it is T1 and normal.

That is, there is a property called either T4 or normal. While we can study spaces which have that property alone, it is usually more interesting to study spaces which have that property and the T1 property. Such spaces are called normal or T4, respectively, depending on what name we assigned to the property. That’s the rub: is the property itself called T4 or normal? Then the other term is used for the combination with T1.

I choose to use the terminology typified by

a topological space is normal iff it is T1 and T4

and I will explain why after we’ve seen the definitions. (This is the terminology of Steen & Seebach, Munkres, and Sieradski. The other terminology is used by Willard, Kasriel, McCarty, Kelley, and Adamson; Dugundji assumes that all topological spaces are T2 = Hausdorff, in which case the two terminologies coincide.) Based on simply counting authors on each side, I should go with the other convention. But I’m going to go with my favorite topology author, Munkres, and I have another reason for my choice.

There is an interesting wiki article about the history of the terminology here. This is what sent me off to get Steen & Seebach and Willard. To judge from the wiki article, the terminology I have chosen is old-fashioned, but not entirely out of fashion. These two books do, in fact, seem to epitomize the two alternative terminologies, because they are both thorough and general.

Let me remind you that a topology is a very general object. Willard puts it very nicely (p. 23):

“The ‘reasonableness’ of the following definition… can best be justified only by reading the forty-two sections which follow it.”

Given a set X, any collection of subsets T (called open sets) is called a topology for X if

  • the empty set \phi\ , and X are in T
  • the arbitrary union of elements of T is in T
  • the finite intersection of elements of T is in T.

The pair (X, T) is called a topological space.

The two extreme examples of a topology are

  • the discrete: T = {every subset of X} = the power set of X.
  • the indiscrete: T = \{\phi, X\}\ .

(So the discrete topology is the largest possible topology on a set X and the indiscrete is the smallest possible.)

An intermediate example is:

Let X = {a, b, c}, a set of 3 elements. Let T = \{ \phi, X, \{a\}, \{b,\ c\} \}\ . That is, the singleton {a} is open but {b} and {c} are not; {b, c} is open but neither {a, b} nor {a, c} is open.

The point is that a topology is any collection we want, subject only to the three somewhat loose requirements.

I will also use that intermediate example later.

For his introduction to the separation axioms, Willard is again worth quoting (p. 85):

“Our definition of a topology admits structures which are, for most purposes, useless.”

Considering that Dugundji assumes that all spaces are Hausdorff (T2), I used to think that the definition of a topological space was a case – a rare case – of too much generalization, of a definition beautiful and simple but too general to be useful. There are, however, at least two kinds of non-Hausdorff topological spaces: the Zariski topology on an algebraic variety (it is T1), and the quotient topology on a space of equivalence classes (it need not even be T0). (Folland, “Real Analysis: Modern Techniques and their Applications; Wiley Interscience 1999, 2nd ed.)

First we look at separating points by means of open sets. (The indiscrete topology has the defect that each singleton set or “point” of X is contained in exactly one open set, X itself. We can’t distinguish any points from each other by using the open sets, i.e. by using the topology. Hence the name.)

T0 property and spaces

A topological space X has the T0 property if there exists an open set which separates any two distinct points: if x and y are distinct points of X, there exist an open set which contains one but not the other.

Let me be more explicit. A topological space X has the T0 property if, for any two distinct points x and y in X, either there exists an open set M(x) containing x which does not contain y, or there exists an open set N(y) containing y which does not contain x.

NOTE that the space X is an open set containing x, but it contains y, and vice versa. We are asserting the existence of a smaller open set, but not necessarily for both points of the pair. If we assert that both M(x) and N(y) exist, that’s the next property.

Here’s a picture of T0, showing an open set containing y that does not contain x. A T0 space is sometimes, but rarely I think, called Kolmogorov.

t0

T1 property and spaces

A topological space X has the T1 property if x and y are distinct points of X, there exists an open set M(x) which contains x but not y, and an open set N(y) which contains y but not x.

One crucial property of a T1 space is that points (singleton sets) are closed.

This time each point has an open set which contains it but not the other. NOTE that we did not assert that the two open sets do not intersect, merely that their intersection contains neither x nor y. (That’s the next property.) Here’s a picture of T1, showing open sets which intersect, but their intersection, as we require, does not contain x or y. A T1 space is sometimes, but again rarely, I think, called Frechet.

t1

T2 property and spaces

A topological space X has the T2 property if x and y are distinct points of X, there exist disjoint open sets M(x) and N(y) containing x and y respectively. Here’s a picture of T2. A T2 space is almost always, in my experience, called Hausdorff.

One crucial property of a Hausdorff space is that limit points are unique. (No, I haven’t defined a limit point. That’s another interesting subject.)

t2

T2 1/2 property and spaces

A topological space X has the T2 1/2 property if there exist open sets whose closures are disjoint, which contain any two distinct points: for any distinct points x and y, there exist open sets A = M(x) and B = N(y) such that \bar{A} \cap \bar{B} = \phi\ . I have seen this called either completely Hausdorff (Steen & Seebach) or Urysohn (Willard); the problem is that both those names are used for two different things (see below). I have only seen the “T2 1/2” used to denote the property I have defined, but I’m not that widely read. Because of the inconsistency, I’m going to refer to T2 1/2 in what follows, although this does not seem to be widely used. I may adopt it permanently for myself, until and unless it leads to difficulties.

Here’s a picture, using solid lines to indicate the closures of the open sets containing x and y.

t2-5

To this point, we have that

T2\ 1/2 \Rightarrow T2 \Rightarrow T1 \Rightarrow T0\ .

Life is good.

T3 and regular

Now we look at separating sets instead of points, still separating them by open sets of some kind.

First we separate a point and a closed set. (A set A in X is closed if its complement X – A is open; the closure of A, \bar{A}\ , is the smallest closed set containing A.)

A topological space X has the T3 property if there exist disjoint open sets which contain any closed set and any point not in the set: for any closed set B and any point x \notin B, there exist disjoint open sets containing x and B respectively.

Here’s T3. This time I use uppercase (“B”) and color to denote the closed set.

t3

It is crucial that the following set and topology (shown earlier as “an intermediate example”) is T3 but not T1 (the problem is that the point a is not closed):

X = {a, b, c}
T = \{ \phi,\ X,\ \{a\},\ \{b,\ c\} \}\ .

This is why and where we need to combine properties in order to get especially worthwhile topological spaces. (Yes, we can study T3, T4, and T5 spaces per se. it is more fruitful to study T3 + T1, T4 + T1, and T5 + T1.)

We say that a space is regular if it is T1 and T3.

(In fact, we can show that if a space is T0 and T3, then it is T2, hence T1, hence T1 and T3. this means we could have defined a space as regular if it is T0 and T3. Of course, T1 and T3 immediately implies T0 and T3, so the two possible definitions of “regular” are equivalent.)

Although I used “normal” and “T4” in the introductory discussion, the alternative terminology appears here as well. It applies to all subscripts 3 and higher. Where I say that a topological space is

regular iff it is T1 and T3

other people use regular to refer to my T3 property, and say a topological space is

T3 iff T1 and regular.

Whereas the progression of the earlier separation axioms kept tightening the requirements on the open sets whose existence we asserted, here we just replaced a point by a closed set. That would be a refinement of the earlier property if points themselves were closed sets. But that’s T1, and that’s why we want to study spaces which are both T1 and T3.

T4 and normal

Now we separate two closed sets instead of a point and a closed set. A topological space X has the T4 property if there exist disjoint open sets which contain any two disjoint closed sets: for any disjoint closed sets A and B, there exist disjoint open sets containing A and B respectively.

t4b

I should mention that a bad property of T4 spaces is that T4 is not hereditary: not every subspace of T4 is T4.

We say that a space is normal if it is T1 and T4. We still have the analogous: not very subspace of a normal space is normal.

T5 and completely normal

Two subsets A and B of topological space are separated if

A \cap \bar{B} = \phi = \bar{A} \cap B\ .

A topological space X has the T5 property if there exist disjoint open sets which contain any two separated sets: for any separated sets A and B, there exist disjoint open sets containing A and B respectively.

t5b

I should mention that an alternative equivalent definition of T5 is that: a space is T5 iff every subspace is T4. It corrects the problem with T4.

We say that a space is completely normal if it is T5 and T1. We have the analogous: a space is completely normal iff every subspace is normal. It corrects the problem with normal, too.

Consider the two open intervals A = (0, 1/2) and B = (1/2,1) with the usual topology of the real line. The sets do not intersect: A \cap B = \phi\ ,, but the closed intervals, their closures, do:
\bar{A} = [0,\ 1/2]
\bar{B} = [1/2,\ 1]
and \bar{A} \cap \bar{B} = \{1/2\}\ .
Nevertheless, A and B are separated, because A \cap \bar{B} = \phi = \bar{A} \cap B\ .

A and B have the T5 property because A and B themselves are disjoint open sets.

All of those properties, T0 thru T5, asserted the existence of open sets, sometimes satisfying additional conditions.

T3 1/2 and completely regular

We have an intermediate property which is described differently.

Given two disjoint subsets A and B of a space X, a Urysohn function for A and B is a continuous function f:X \longrightarrow [0,1] such that f(A) = 0 and f(B) = 1.

Urysohn’s Lemma, then, says that if A and B are disjoint closed subsets of a T4 space, then there exists a Urysohn function for A and B.

A topological space X has the T3 1/2 property if there exist a real-valued continuous function which separates an open set from any point not in it: i.e. for each open set U \subset X and each x not in U, there exist a Urysohn function f for x and U.

We say that a space is completely regular (or Tychonoff) if it is T3 1/2 and T1.

Urysohn function for distinct points

We could, instead, require this kind of property for two points: we would say that a space is Urysohn if it has a Urysohn function for any two distinct points.

Implications of the properties

At this point, thanks to adding T1 to the definitions, we can show (!)

completely normal \Rightarrow\ normal \Rightarrow\ completely regular \Rightarrow\ regular \Rightarrow\ T2 1/2 \Rightarrow\ T2 \Rightarrow\ T1 \Rightarrow\ T0.

The implications among the Ti properties (for i > 2 1/2) are not so pretty.

Note that a Urysohn space was not in that list. Instead of the subsequence

completely regular \Rightarrow\ regular \Rightarrow\ T2 1/2

we could have written

completely regular \Rightarrow\ Urysohn \Rightarrow\ T2 1/2.

but there is no inclusion relationship between Urysohn and regular. We have two beautiful inclusions, if we omit either regular or Urysohn, but not if we include both.

This is the second reason why I decided to follow Steen & Seebach and use T’s for the properties and names for the combinations. If we did it the other way, with names for the properties and T’s for the combinations, we could write

T5 \Rightarrow\ T4 \Rightarrow\ T31/2 \Rightarrow\ T3 \Rightarrow\ T2 1/2 \Rightarrow\ T2 \Rightarrow\ T1 \Rightarrow\ T0,

or, more elegantly,

Ti \Rightarrow\ Tj for i > j, with i, j in {0,1,2,21/2,3, 31/2,4,5}

but then we’ve left Urysohn spaces out in the cold. Since the theorem is no longer pretty, I chose to use the shorter Ti to denote a property, and write, for example,

normal = T1 + T4.

I first saw them the other way:

T4 = normal + T1, etc.

and it is possible that I would not have been so struck by them without the lovely Ti \Rightarrow\ Tj for i > j. (Adamson emphasizes that he chooses this convention because of the simplicity of that statement.) Nevertheless, I have presented them the other way.

The fact is, if you’re studying someone else’s work, you may have to adopt their terminology as long as you’re there.

One final caveat: completely hausdorff and Urysohn

Willard and Steen & Seebach use exactly opposite definitions for completely Hausdorff and Urysohn. Willard says a space is completely Hausdorff if any two distinct points have a Urysohn function, while a space is Urysohn if any two distinct points are separated by open sets whose closures are disjoint (T2 1/2).

It seems odd to say that a completely Hausdorff space is separated by a Urysohn function, while a Urysohn space is defined by separation by sets.

Note that this naming mess is between Urysohn and completely Hausdorff, while the two spaces that mess up our nice inclusion are regular (“T 2 1/2”) and Urysohn (“two distinct points have a Urysohn function”). Note, also, that unlike the confusion between T4 and normal, this mess has nothing to do with T1.

My own inclination, on this, is to use T2 1/2 for the space defined by separation by sets, and Urysohn for the space defined by “two distinct points have a Urysohn function”.

Ah, let me close with a few remarks. First, every metric space – with the usual topology defined by open balls – is completely normal (T1 + T5). Note that I have specified my naming convention. The implication does not go the other way: normality is not sufficient to imply metrizability (that the topology can be gotten from a metric on X).

Second, there are a host of fascinating properties which I have not mentioned: first and second countable, first and second category, assorted forms of connectedness, and assorted forms of compactness.

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