# Existence of Topological Space which satisfies no Separation Axioms but T0 and T1

## Theorem

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom and $T_1$ (Fréchet) axiom.

## Proof

Let $T$ be the finite complement topology on a countable space.

From Finite Complement Space is T1, $T$ is a $T_1$ (Fréchet) space.

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

The rest of the result follows from:

Finite Complement Space is not $T_2$

and:

Finite Complement Space is not $T_3$, $T_4$ or $T_5$

$\blacksquare$