Existence of Hausdorff Space which is not T3, T4 or T5
Jump to navigation
Jump to search
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:
$\blacksquare$
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