R. Smith

will speak on

The continuity of betweenness

Time: 4:00PM
Date: Fri 27th October 2017
Location: UCD Science North 125 [map]

Abstract: Given a set $X$, we can use a suitable ternary relation
$[\cdot,\cdot,\cdot] \subseteq X^3$ to express the notion
of `betweenness' on $X$: $x$ is between $a$ and $b$ if and
only if $[a,x,b]$ holds. We assume that this relation is
"basic": $[a,a,b]$ and $[a,b,b]$ always hold, $[a,x,b]$
implies $[b,x,a]$, and $[a,x,a]$ implies $x=a$. Many natural
examples of betweenness arise when $X$ is endowed with some
additional order-theoretic or topological structure. Given
$a,b \in X$, we can define the "interval" $[a,b] =
\lbrace x \in X\,:\,[a,x,b]\rbrace\;(= [b,a])$. If $X$ has additional
topological structure, it is reasonable to ask whether the
assignment $\lbrace a,b\rbrace \mapsto [a,b]$ has good continuity
properties, given a suitable hyperspace topology. We examine
this question in the context of "Menger betweenness"
on metric spaces $(X,d)$ ($[a,x,b]$ holds if and only if
$d(a,b)=d(a,x)+d(x,b)$), and the "K-interpretation
of betweenness" on topological continua ($[a,x,b]$ holds
if and only if $x$ is an element of every subcontinuum that
includes $a$ and $b$).

(This talk is part of the Analysis series.)

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