Section Separation Axioms
As we have seen, sequences in topological spaces do not generally behave as we would expect them to. As a result, we look for conditions on topological spaces under which sequences do exhibit some regular behavior. In our preview activity we saw that it is possible in a topological space that single point sets do not have to be closed. In
Activity 13.4, we also saw that limits of sequences in topological spaces are not necessarily unique. This type of behavior limits the results that one can prove about such spaces. As a result, we define classes of topological spaces whose behaviors are closer to what our intuition suggests.
Activity 13.7.
(a)
Consider a metric space
and let
and
be distinct points in
(i)
Explain why
and
cannot both be limits of the same sequence if we can find disjoint open balls
centered at
and
centered at
such that
(ii)
Now show that we can find disjoint open balls
centered at
and
centered at
such that
(b)
We saw that every point in
is a limit of the constant sequence
If
in
Explain why there are no disjoint open sets
containing
and
containing
It is the fact as described in
Activity 13.7 that we can separate disjoint points by disjoint open sets that separates metric spaces from other spaces where limits are not unique. If we restrict ourselves to spaces where we can separate points like this, then we might expect to have unique limits. Such spaces are called
Hausdorff spaces.
Definition 13.12.
A topological space
is a
Hausdorff space if for each pair
of distinct points in
there exists open sets
of
and
of
such that
Activity 13.7 shows that every metric space is a Hausdorff space. Once we start imposing conditions on topological spaces, we restrict the number of spaces we consider.
Activity 13.8.
(a)
Let
be any set and
the discrete topology. Is
Hausdorff? Justify your answer.
(b)
Let
be a Hausdorff topological space with
a finite set. Is
an open set? Explain. What does this say about the topology
Example 13.13.
There are examples of Hausdorff spaces that are not the standard metric spaces. For example, Let
We use
to make a topology on
with basis all open intervals of the form
and all sets of the form
where
are real numbers. This topology adds the extra intervals of the form
to the standard open intervals to make a new topology. This topology is known as the
-topology on
Just as in
if
and
are distinct real numbers we can separate
and
with open intervals.
The reason we defined Hausdorff spaces is because they have familiar properties, as the next theorems illustrate.
Theorem 13.14.
Each single point subset of a Hausdorff topological space is closed.
Proof.
Let
be a Hausdorff topological space, and let
for some
To show that
is closed, we prove that
is open. Let
Then
So there exist open sets
containing
and
containing
such that
So
and
Thus, every point of
is an interior point and
is an open set. This verifies that
is a closed set.
Theorem 13.15.
A sequence of points in a Hausdorff topological space can have at most one limit in the space.
Proof.
Let
be a Hausdorff topological space, and let
be a sequence in
Suppose
converges to
and to
Suppose
Then there exist open sets
of
and
of
such that
But the fact that
converges to
implies that there is a positive integer
such that
for all
But then
for any
This contradicts the fact that
converges to
We conclude that
and that the sequence
can have at most one limit in
Hausdorff spaces are important because we can separate distinct points with disjoint open sets. It is also of interest to consider what other types of objects we can separate with disjoint open sets. For example, the indiscrete topology is quite bad in the sense that its open sets canโt distinguish between distinct points. That is, if
and
are distinct points in a space with the indiscrete topology, then every open set that contains
also contains
By contrast, in a Hausdorff space we can separate distinct points with disjoint open sets. This is an example of what is called a โseparationโ property. Other types of separation properties describe different types of topological spaces. These separation properties determine what kind of objects we can separate with disjoint open sets โ e.g., points, points and closed sets, closed sets and closed sets. The following are the most widely used separation properties. These properties rule out kinds of unwelcome properties that a topological space might have. For example, recall that limits of sequences are unique in Hausdorff spaces. (We traditionally call these separation properties โaxiomsโ because we generally assume that our topological spaces have these properties. However, these are not axioms in the usual sense of the word, but rather properties.)
Definition 13.16.
Let
be a topological space.
The space
is a
T-space or
Frechet space if for every
in
there exist an open set
containing
such that
The space
is a
T-space or a
Hausdorff space if for every
in
there exist disjoint open sets
and
such that
and
The space
is
regular if for each closed set
of
and each point
there exists disjoint open sets
and
in
such that
and
The space
is a
T-space or a
regular Hausdorff space if
is a regular
space.
The space
is a
normal space if for each pair
and
of disjoint closed subsets of
there exist disjoint open sets
and
such that
and
The space
is a
T-space or a
normal Hausdorff space if
is a normal
space.
Exercise 16 shows that every metric space is both regular and normal. The use of the variable
comes from the German โTrennungsaxiomeโ for separation axioms. Note again that these are not technically axioms, but rather properties. An interesting question is why we insist that
and
-spaces also be
We want these axioms to provide more separation as the index increases. Consider a space
with the indiscrete topology. In this space, nothing is separated. However, this space is vacuously regular and normal. To avoid this seeming incongruity, we insist on working only with
spaces. Note that a space with the indiscrete topology is not
It is the case that every
-space is
every
-space is
and every
-space is
Verification of these statements are left to
Exercise 18. These properties are also all different. That is, there are
-spaces that are not
and
-spaces that are not
These problems are given in
Exercise 19. The fact that there are
-spaces that are not
is a bit difficult. An example is the
Niemytzki plane. The Niemytzki plane is the upper half plane
Let
be the boundary of
That is,
A basis for the topology on
consists of the standard open disks centered at points with
along with the open disks in
that are tangent to
together with their points of tangency. We wonโt verify that the Niemytzki plane is
but not
The interested reader can find an accessible proof in the article โAnother Proof that the Niemytzki Plane is not Normalโ by David H. Vetterlein in the
Pi Mu Epsilon Journal, Vol. 10, No. 2 (SPRING 1995), pp. 119-121.