Definition:Double Pointed Topology

Definition
Let $T = \left({S, \vartheta}\right)$ be a topological space.

Let $S' = \left\{{x, y}\right\}$ be a doubleton.

Let $D = \left({S', \left\{{\varnothing, S'}\right\}}\right)$ be the indiscrete space on $D$.

Let $T \times D$ 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

 * A Double Pointed Topology is Not $T_0$.