Definition:Double Pointed Topology

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.


 * Double Pointed Topology is not $T_0$