# Separation Axioms on Double Pointed Topology/T3.5 Axiom

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

## Proof

## 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: $1$