Definition:Nested Interval Topology

Definition

Let $S = \openint 0 1$ denote the open real interval:

$\openint 0 1 = \set {x \in \R: 0 < x < 1}$

Let $\tau$ be the topology defined on $\openint 0 1$ by defining the open sets $U_n$ as:

$\forall n \in \N_{>0}: U_n := \openint 0 {1 - \dfrac 1 n}$

together with $\O$ and $S$ itself.

Then $\tau$ is referred to as the nested interval topology.

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

Also see

• Nested Interval Topology is Topology

• Existence of Weakly Sigma-Locally Compact Space which is not Strongly Locally Compact

• Results about the nested interval topology can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $52$. Nested Interval Topology