Definition:Concentric Circle Topology

Definition

Open Set of $T$

Let $C_1$ and $C_2$ be concentric circles in the Cartesian plane $\R^2$ such that $C_1$ is inside $C_2$.

Let $S = C_1 \cup C_2$.

Let $\BB$ be the set of sets consisting of:

all singleton sets of $C_2$

all open intervals on $C_1$ each together with its projection from the center of the circles onto $C_2$ except for the midpoint.

$\BB$ is then taken to be the sub-basis for a topology $\tau$ on $S$.

Thus $\tau$ is referred to as the concentric circle topology.

The topological space $T = \struct {S, \tau}$ is referred to as the concentric circle space.

Also see

• Results about the concentric circle topology can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $97$. Concentric Circles