Interval (Topology)

Definition

Let $(X,\le )$ be a total order. Let $\overline{X}=X\cup \{-\infty ,+\infty \}$ be the extended set where $-\infty <x<+\infty$ for all $x\in X$. For $a,b\in \overline{X}$, we define the standard intervals as subsets of $X$ determined by their relationship to these endpoints.

Interval Types

For $a,b\in X$:

• Closed: $[a,b]=\{x\in X\mid a\le x\le b\}$
• Half-open: $[a,b)=\{x\in X\mid a\le x<b\}$ and $(a,b]=\{x\in X\mid a<x\le b\}$

For $a,b\in \overline{X}$:

• Open: $(a,b)=\{x\in X\mid a<x<b\}$

For infinite bounds specifically:

• $(-\infty ,b]=\{x\in X\mid x\le b\}$ for $b\in X$
• $[a,+\infty )=\{x\in X\mid a\le x\}$ for $a\in X$
• $(-\infty ,+\infty )=X$

Since $\pm \infty \notin X$, we restrict the use of closed brackets to endpoints in $X$ to ensure the interval remains a subset of $X$.

Convexity

A subset $I\subseteq X$ is an interval if and only if it is convex in $X$.

Definition: Convex Set

A subset $I\subseteq X$ is convex iff: $\forall x,y\in I,\forall z\in X,(x<z<y⟹z\in I)$

Edge Cases and Degeneracy

1. Empty Set: If $a>b$ in $\overline{X}$, then $(a,b)=\varnothing$.
2. Singletons: For $a\in X$, $[a,a]=\{a\}$, while $(a,a)=\varnothing$.
3. Infinite Bounds: Because $\pm \infty \notin X$, the notation $[-\infty ,b]$ is formally equivalent to $(-\infty ,b]$.
4. Intersections: The intersection of any collection of intervals is an interval.

References

• Total Ordering
• Order Topology
• Dedekind-MacNeille Completion