Existence of Topological Space which satisfies no Separation Axioms but T0 and T1
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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 $T_1$, $T$ 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:
and:
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms