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Topology: An Inquiry-Based Approach

Steven Schlicker

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 \((X,d)\text{,}\) and let \(x\) and \(y\) be distinct points in \(X\text{.}\)
(i)
Explain why \(x\) and \(y\) cannot both be limits of the same sequence if we can find disjoint open balls \(B(x,r)\) centered at \(x\) and \(B(y,s)\) centered at \(y\) such that \(B_x \cap B_y = \emptyset\text{.}\)
(ii)
Now show that we can find disjoint open balls \(B(x,r)\) centered at \(x\) and \(B(y,s)\) centered at \(y\) such that \(B(x,s) \cap B(y,r) = \emptyset\text{.}\)

(b)

Return to our example from Activity 13.4 with \(X = \{a, b, c\}\) and topology
\begin{equation*} \tau~=~\{\emptyset, \{c\}, \{a, c\}, \{b, c\}, X\}\text{.} \end{equation*}
We saw that every point in \(X\) is a limit of the constant sequence \((c)\text{.}\) If \(x \neq c\) in \(X\text{,}\) Explain why there are no disjoint open sets \(O_x\) containing \(x\) and \(O_c\) containing \(c\text{.}\)
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 \(X\) is a Hausdorff space if for each pair \(x\text{,}\) \(y\) of distinct points in \(X\text{,}\) there exists open sets \(O_x\) of \(x\) and \(O_y\) of \(y\) such that \(O_x \cap O_y = \emptyset\text{.}\)
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 \(X\) be any set and \(\tau\) the discrete topology. Is \((X, \tau)\) Hausdorff? Justify your answer.

(b)

Let \((X, \tau)\) be a Hausdorff topological space with \(X = \{x, x_1, x_2, \ldots, x_n\}\) a finite set. Is \(\{x\}\) an open set? Explain. What does this say about the topology \(\tau\text{?}\)
Hint.
Is \(x\) a limit point of \(\{x_1, x_2, \ldots, x_n\}\text{?}\)

Example 13.13.

There are examples of Hausdorff spaces that are not the standard metric spaces. For example, Let \(K = \left\{\frac{1}{k} \mid k \text{ is a positive integer } \right\}\text{.}\) We use \(K\) to make a topology on \(\R\) with basis all open intervals of the form \((a,b)\) and all sets of the form \((a,b) \setminus K\text{,}\) where \(a \lt b\) are real numbers. This topology adds the extra intervals of the form \((a,b) \setminus K\) to the standard open intervals to make a new topology. This topology is known as the \(K\)-topology on \(\R\text{.}\) Just as in \((\R, d_E)\text{,}\) if \(x\) and \(y\) are distinct real numbers we can separate \(x\) and \(y\) with open intervals.
The reason we defined Hausdorff spaces is because they have familiar properties, as the next theorems illustrate.

Proof.

Let \(X\) be a Hausdorff topological space, and let \(A = \{a\}\) for some \(a \in X\text{.}\) To show that \(A\) is closed, we prove that \(X \setminus A\) is open. Let \(x \in X \setminus A\text{.}\) Then \(x \neq a\text{.}\) So there exist open sets \(O_x\) containing \(x\) and \(O_a\) containing \(a\) such that \(O_x \cap O_a = \emptyset\text{.}\) So \(a \notin O_x\) and \(O_x \subseteq X \setminus A\text{.}\) Thus, every point of \(X \setminus A\) is an interior point and \(X \setminus A\) is an open set. This verifies that \(A\) is a closed set.

Proof.

Let \(X\) be a Hausdorff topological space, and let \((x_n)\) be a sequence in \(X\text{.}\) Suppose \((x_n)\) converges to \(a \in X\) and to \(b \in X\text{.}\) Suppose \(a \neq b\text{.}\) Then there exist open sets \(O_a\) of \(a\) and \(O_b\) of \(b\) such that \(O_a \cap O_b = \emptyset\text{.}\) But the fact that \((x_n)\) converges to \(a\) implies that there is a positive integer \(N\) such that \(x_n \in O_a\) for all \(n \geq N\text{.}\) But then \(x_n \notin O_b\) for any \(n \geq N\text{.}\) This contradicts the fact that \((x_n)\) converges to \(b\text{.}\) We conclude that \(a=b\) and that the sequence \((x_n)\) can have at most one limit in \(X\text{.}\)
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 can’t distinguish between distinct points. That is, if \(x\) and \(y\) are distinct points in a space with the indiscrete topology, then every open set that contains \(x\) also contains \(y\text{.}\) By contrast, in a Hausdorff space we can separate distinct points with disjoint open sets. This is an example of what is called a “separation” 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 — 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 “axioms” 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 \(X\) be a topological space.
  1. The space \(X\) is a T\(_1\)-space or Frechet space if for every \(x\neq y\) in \(X\text{,}\) there exist an open set \(U\) containing \(y\) such that \(x \notin U\text{.}\)
  2. The space \(X\) is a T\(_2\)-space or a Hausdorff space if for every \(x\neq y\) in \(X\text{,}\) there exist disjoint open sets \(U\) and \(V\) such that \(x\in U\) and \(y\in V\text{.}\)
  3. The space \(X\) is regular if for each closed set \(C\) of \(X\) and each point \(x \in X \setminus C\text{,}\) there exists disjoint open sets \(U\) and \(V\) in \(X\) such that \(C \subseteq U\) and \(x \in V\text{.}\) The space \(X\) is a T\(_3\)-space or a regular Hausdorff space if \(X\) is a regular \(T_1\) space.
  4. The space \(X\) is a normal space if for each pair \(C\) and \(D\) of disjoint closed subsets of \(X\) there exist disjoint open sets \(U\) and \(V\) such that \(C \subseteq U\) and \(D \subseteq V\text{.}\) The space \(X\) is a T\(_4\)-space or a normal Hausdorff space if \(X\) is a normal \(T_1\) space.
Exercise 16 shows that every metric space is both regular and normal. The use of the variable \(T\) comes from the German “Trennungsaxiome” for separation axioms. Note again that these are not technically axioms, but rather properties. An interesting question is why we insist that \(T_3\) and \(T_4\)-spaces also be \(T_1\text{.}\) We want these axioms to provide more separation as the index increases. Consider a space \(X\) 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 \(T_1\) spaces. Note that a space with the indiscrete topology is not \(T_1\text{.}\)
It is the case that every \(T_4\)-space is \(T_3\text{,}\) every \(T_3\)-space is \(T_2\text{,}\) and every \(T_2\)-space is \(T_1\text{.}\) Verification of these statements are left to Exercise 18. These properties are also all different. That is, there are \(T_1\)-spaces that are not \(T_2\) and \(T_2\)-spaces that are not \(T_3\text{.}\) These problems are given in Exercise 19. The fact that there are \(T_3\)-spaces that are not \(T_4\) is a bit difficult. An example is the Niemytzki plane. The Niemytzki plane is the upper half plane \(X = \{(x,y) \in \R^2 \mid y \geq 0\}\text{.}\) Let \(L\) be the boundary of \(X\text{.}\) That is, \(L = \{(x,y) \in \R^2 \mid y = 0\}\text{.}\) A basis for the topology on \(X\) consists of the standard open disks centered at points with \(y \gt 0\) along with the open disks in \(X \setminus L\) that are tangent to \(L\) together with their points of tangency. We won’t verify that the Niemytzki plane is \(T_3\) but not \(T_4\text{.}\) The interested reader can find an accessible proof in the article “Another Proof that the Niemytzki Plane is not Normal” by David H. Vetterlein in the Pi Mu Epsilon Journal, Vol. 10, No. 2 (SPRING 1995), pp. 119-121.
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