# A T2 Space That Is NOT a T3 Space

Recall from the T3 (Regular Hausdorff) Topological Spaces page that a topological space $X$ is said to be regular if for all $x \in X$ and all closed sets $F$ of $X$ not containing $x$ there exists open sets $U$ and $V$ that separate $\{ x \}$ and $F$.

We then said that a topological space $X$ is a T_{3} space if it is both a regular space and a T_{1} space.

We saw that every T_{3} space is a T_{2} space. We will now see that there exists T_{2} spaces that are not T_{3} spaces.

Consider the following half-plane:

(1)Define the topology on $X$ whose basis elements are open disks contained in $X$:

As well as sets which we denote by:

(2)Then $X$ with this topology is clearly T_{2}. Any two points not on the $x$-axis can be separated by open balls. A point not on the $x$-axis can be separated from a point on the $x$-axis by taking an open ball and open half-disk with a sufficiently small radius. The same goes for two points on the $x$-axis.

However, we claim that $X$ is not a T_{3} space. Consider the following set:

Take the point $\mathbf{x} = (1, 0)$. Then any open set containing $\mathbf{x}$ will contain a set of the form $U_{1, r}$ where $r > 0$. But since $r \neq 0$, $U_{1, r}$ will intersect any open set containing $F$ and hence $X$ is not a T_{3} space.