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: $1$