# Double Pointed Topology is not T0

## Theorem

Let $T_1 = \struct {S, \tau_S}$ be a topological space.

Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$.

Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$.

Then $T$ is not a $T_0$ (Kolmogorov) space.

## Proof 1

By definition, the double pointed topology $\tau$ on $T_1$ is the product topology on $T_1 \times D$.

Let $x \in S$, and consider the point $\tuple {x, a} \in S \times A$.

Then:

- $\forall U \in \tau: \tuple {x, a} \in U \implies \tuple {x, b} \in U$
- $\forall U \in \tau: \tuple {x, b} \in U \implies \tuple {x, a} \in U$

as $D$ is an indiscrete space.

Hence the result, by definition of $T_0$ (Kolmogorov) space.

$\blacksquare$

## Proof 2

By definition, the double pointed topology $\tau$ on $T_1$ is the product topology on $T_1 \times D$.

By definition, $D$ is the indiscrete space on a doubleton.

Aiming for a contradiction, suppose $T$ is a $T_0$ (Kolmogorov) space.

Then from Product Space is $T_0$ iff Factor Spaces are $T_0$ it follows that $D$ is also a $T_0$ (Kolmogorov) space.

But from Indiscrete Non-Singleton Space is not $T_0$, $D$ is not a $T_0$ (Kolmogorov) space.

The result follows by Proof by Contradiction.

$\blacksquare$

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Notes: Part $1$: Basic Definitions: Section $2$. Separation Axioms: $1$