Separation Axioms Preserved under Homeomorphism

Theorem

The separation axioms are preserved under homeomorphism.

Let $T_A = \left({S_A, \tau_A}\right), T_B = \left({S_B, \tau_B}\right)$ be topological spaces.

Let $\phi: T_A \to T_B$ be a homeomorphism.

$T_0$ (Kolmogorov) Space is Preserved under Homeomorphism

If $T_A$ is a $T_0$ (Kolmogorov) space, then so is $T_B$.

$T_1$ (Fréchet) Space is Preserved under Homeomorphism

If $T_A$ is a $T_1$ (Fréchet) space, then so is $T_B$.

Hausdorff Condition is Preserved under Homeomorphism

If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$.

$T_{2 \frac 1 2}$ (Completely Hausdorff) Space is Preserved under Homeomorphism

If $T_A$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space, then so is $T_B$.

$T_3$ Space is Preserved under Homeomorphism

If $T_A$ is a $T_3$ space, then so is $T_B$.

Regular Space is Preserved under Homeomorphism

If $T_A$ is a regular space, then so is $T_B$.

$T_{3 \frac 1 2}$ Space is Preserved under Homeomorphism

If $T_A$ is a $T_{3 \frac 1 2}$ space, then so is $T_B$.

Tychonoff (Completely Regular) Space is Preserved under Homeomorphism

If $T_A$ is a Tychonoff (completely regular) space, then so is $T_B$.

$T_4$ Space is Preserved under Homeomorphism

If $T_A$ is a $T_4$ space, then so is $T_B$.

Normal Space is Preserved under Homeomorphism

If $T_A$ is a normal space, then so is $T_B$.

$T_5$ Space is Preserved under Homeomorphism

If $T_A$ is a $T_5$ space, then so is $T_B$.

Completely Normal Space is Preserved under Homeomorphism

If $T_A$ is a completely normal space, then so is $T_B$.

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: Functions, Products, and Subspaces