# Existence of Hausdorff Space which is not T3, T4 or T5

## Theorem

There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space (and hence also $T_0$ (Kolmogorov) and $T_1$ (Fréchet), but is not a $T_3$ space, a $T_4$ space or a $T_5$ space.

## Proof

Let $T$ be an irrational slope topological space.

From Irrational Slope Space is $T_2$, $T$ is a $T_2$ (Hausdorff) space.

Hence from $T_2$ Space is $T_1$ Space and $T_1$ Space is $T_0$ Space, also a $T_0$ (Kolmogorov) space and a $T_1$ (Fréchet) space.

The rest of the result follows from:

Irrational Slope Space is not $T_3$, $T_4$ or $T_5$.

$\blacksquare$