Separation Axioms on Double Pointed Topology

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Theorem

Let $T = \struct {S, \tau_S}$ be a topological space.

Let $D = \struct {\set {a, b}, \tau_D}$ be the indiscrete topology on two points.

Let $T \times D$ be the double pointed topology on $T$.


Then:

$T \times D$ is not a $T_0$ (Kolmogorov) space, a $T_1$ (Fréchet) space, a $T_2$ (Hausdorff) space or a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
$T \times D$ is a $T_3$ space, a $T_{3 \frac 1 2}$ space, a $T_4$ space or a $T_5$ space if and only if $T$ is.


Proof

$T_0$, $T_1$, $T_2$ and $T_{2 \frac 1 2}$ Axioms

From:

Double Pointed Topology is not $T_0$
$T_1$ Space is $T_0$ Space
$T_2$ Space is $T_1$ Space
Completely Hausdorff Space is $T_2$ Space

we have that $T \times D$ is not $T_0$, $T_1$ or $T_2$ or $T_{2 \frac 1 2}$.

$\Box$


From:

Separation Axioms on Double Pointed Topology: $T_3$ Axiom

$T \times D$ is a $T_3$ space if and only if $T$ is.

$\Box$


From:

Separation Axioms on Double Pointed Topology: $T_{3 \frac 1 2}$ Axiom

$T \times D$ is a $T_{3 \frac 1 2}$ space if and only if $T$ is.

$\Box$


From:

Separation Axioms on Double Pointed Topology: $T_4$ Axiom

$T \times D$ is a $T_4$ space if and only if $T$ is.

$\Box$


From:

Separation Axioms on Double Pointed Topology: $T_5$ Axiom

$T \times D$ is a $T_5$ space if and only if $T$ is.

$\blacksquare$


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