# Definition:Double Pointed Topology

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## Definition

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

Let $A = \set {a, b}$ be a doubleton.

Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on $A$.

Let $\struct {T \times D, \tau}$ be the product space of $T$ and $D$.

Then $T \times D$ is known as the **double pointed topology** on $T$.

It is seen that $T \times D$ is conceptually equivalent to taking the space $T$ and replacing each point with a pair of topologically indistinguishable points.

## Also see

- Definition:Multiple Pointed Topology, of which it can be seen the double pointed topology is a special case.

- Results about
**double pointed topologies**can be found here.

## Sources

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