Hausdorff Topological Spaces Examples 3

Hausdorff Topological Spaces Examples 3

Recall from the Hausdorff Topological Spaces page that a topological space $(X, \tau)$ is said to be a Hausdorff space if for every distinct $x, y \in X$ there exists open neighbourhoods $U, V \in \tau$ such that $x \in U$, $y \in V$ and $U \cap V = \emptyset$.

We also looked at two notable examples of Hausdorff spaces - the first being the set of real numbers $\mathbb{R}$ with the usual topology of open intervals on $\mathbb{R}$, and the second being the discrete topology on any nonempty set $X$.

We will now look at some more problems regarding Hausdorff topological spaces.

Example 1

Prove that every metric space is a Hausdorff space.

Let $X$ be a metric space. The open sets in $X$ are therefore the any set that is the union of a collection of open balls with respect to the metric $d$ defined on $X$.

Let $x, y \in X$ where $x \neq y$. Then $d(x, y) > 0$ and so if we let $r = \frac{d(x, y)}{2}$ then the ball centered at $x$ with radius $r$ and the ball centered at $y$ with radius $r$ are disjoint, that is:

(1)
\begin{align} \quad B(x, r) \cap B(y, r) = \emptyset \end{align}

Furthermore if we set $U = B(x, r)$ and $V = B(y, r)$ we have that $U$ and $V$ are open sets of $X$ with respect to the metric $d$. Therefore any metric space is a Hausdorff space.

Example 2

Prove that if $(X, \tau)$ is a Hausdorff space then for every $x \in X$, the singleton set $\{ x \}$ is closed.

Suppose that $(X, \tau)$ is a Hausdorff space and let $x \in X$. We will show that $X \setminus \{ x \}$ is therefore open. Let $y \in X \setminus \{ x \}$. Since $(X, \tau)$ is Hausdorff there exists open neighbourhoods of $U$ of $x$ and $V$ of $y$ such that $U \cap V = \emptyset$. But since $x \in U$ we must have that $x \not \in V$, so $y \in V \subseteq X \setminus \{ x \}$.

So for every $y \in X \setminus \{ x \}$ there exists a $V \in \tau$ such that $y \in V \subseteq X \setminus \{ x \}$ so $\mathrm{int} (X \setminus \{ x \}) = X \setminus \{ x \}$, i.e., $X \setminus \{ x \}$ is open, and so, $\{ x \}$ is closed.

Example 3

Use Example 2 to show that the set $X = \{ a, b, c, d \}$ with the topology $\tau = \{ \emptyset, \{ a \}, \{a, b \}, \{a, c \}, \{a, b, c \}, X \}$ is not a Hausdorff space.

The contrapositive of the result from Example 2 says that if there exists a singleton set $\{ x \}$ which is open then $(X, \tau)$ is not a Hausdorff space.

With the topology above we see that the singleton set $\{ a \}$ is open. For the element $b \in X$ we see that the only open neighbourhoods of $b$ are $\{ a, b \}$, $\{a, b, c \}$, and $X$. Furthermore, all of the open neighbourhoods of $a$ are $\{a \}$, $\{ a, b \}$, $\{a, c \}$, $\{a, b, c \}$, $X$.

The intersection of any of these open neighbourhoods is nonempty, and so there does not exist any open neighbourhoods $U$ of $a$ and $V$ of $b$ such that $U \cap V = \emptyset$, so $(X, \tau)$ is not a Hausdorff space.