Category:Overlapping Interval Topology

From ProofWiki
Jump to navigation Jump to search

This category contains results about Overlapping Interval Topology.

Let $S = \closedint {-1} 1$ denote the open real interval:

$\closedint {-1} 1 = \set {x \in \R: -1 \le x \le 1}$


Let $\BB$ be the set:

$\BB = \set {\hointl a 1: -1 < a < 0} \cup \set {\hointr {-1} b: 0 < b < 1}$

where:

$\hointl a 1$ is the half-open interval $\set {x \in \R: a < x \le 1}$.
$\hointr {-1} b$ is the half-open interval $\set {x \in \R: -1 \le x < b}$.

Then $\BB$ is the basis for a topology $\tau$ on $\R$.


Thus the sets of the form $\openint a b$ such that $a < 0 < b$ are open sets in $S$.


$\tau$ is referred to as the overlapping interval topology.

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

This category currently contains no pages or media.