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:
$T \times D$ is a $T_3$ space if and only if $T$ is.
$\Box$
From:
$T \times D$ is a $T_{3 \frac 1 2}$ space if and only if $T$ is.
$\Box$
From:
$T \times D$ is a $T_4$ space if and only if $T$ is.
$\Box$
From:
$T \times D$ is a $T_5$ space if and only if $T$ is.
$\blacksquare$
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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: $1$