Definition:Multiple Pointed Topology

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

Let $A$ be a finite set whose cardinality is greater than $1$.

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

Let $T \times D$ be the product space of $T$ and $D$.

Then $T \times D$ is known as the multiple 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 finite set of topologically indistinguishable points.

Also see

 * Double Pointed Topology