Hausdorff Topological Spaces

Definition: Let $(X, \tau)$ be a topological space. If for every distinct $x, y \in X$ there exists open neighbourhoods $U, V \in \tau$ with $x \in U$ and $y \in V$ such that $U \cap V = \emptyset$ then $(X, \tau)$ is said to be a Hausdorff Space.

Essentially $(X, \tau)$ is a Hausdorff space if we can separate each of the elements in $X$ with open neighbourhoods as illustrated below:

Perhaps the most familiar example of a Hausdorff Space is $(\mathbb{R}, \tau)$ where $\tau$ is the usual topology on $\mathbb{R}$ consisting of open intervals. To prove this this topological space is Hausdorff, let $x, y \in X$ with $x \neq y$ and without loss of generality assume that $x < y$. The Euclidean distance between $x$ and $y$ is:

(1)

\begin{align} \quad d(x, y) = \mid x - y \mid \end{align}

Now let $U = \left (x - \frac{d(x, y)}{2}, x + \frac{d(x, y)}{2} \right )$ and $V = \left (y - \frac{d(x, y)}{2}, y + \frac{d(x, y)}{2} \right )$.

Then $U, V \in \tau$, $x \in U$, $y \in V$, and $U \cap V = \emptyset$. But $x, y \in X$ were arbitrary, and so $(\mathbb{R}, \tau)$ is a Hausdorff space.

For another example, let $X$ be any nonempty set and let $\tau$ be the discrete topology on $X$. Then for all $x, y \in X$ we have that $\{ x \}, \{ y \} \in \tau$. If we let $U = \{ x \}$ and $V = \{ y \}$ then $x \in U$, $y \in V$, and $U \cap V = \emptyset$. Since $x, y \in X$ are arbitrary we conclude that $(X, \tau)$ is a Hausdorff Space.