\begin{definition}[Definition:Double Pointed Topology]
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.
\end{definition}