03-00-LineSegment.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{LineSegment}
\pmcreated{2013-03-22 14:19:01}
\pmmodified{2013-03-22 14:19:01}
\pmowner{matte}{1858}
\pmmodifier{matte}{1858}
\pmtitle{line segment}
\pmrecord{12}{35783}
\pmprivacy{1}
\pmauthor{matte}{1858}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{03-00}
\pmclassification{msc}{51-00}
\pmrelated{Interval}
\pmrelated{LinearManifold}
\pmrelated{LineInThePlane}
\pmrelated{CircularSegment}
\pmdefines{open line segment}
\pmdefines{closed line segment}

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\begin{document}
{\bf Definition} 
Suppose $V$ is a vector space over $\sR$ or $\sC$, and $L$ is a subset of $V$. 
Then $L$ is a \emph{line segment} if $L$ can be parametrized
as
$$L = \{ a+tb \mid t\in[0,1]\}$$
for some $a,b$ in $V$ with $b\neq 0$. 

Sometimes one needs to distinguish between open and \PMlinkname{closed}{Closed}
line segments. Then one defines a \emph{closed line segment} 
as above, 
and an \emph{open line segment} as a subset $L$ that can be 
parametrized as
$$L = \{ a+tb \mid t\in(0,1)\}$$
for some $a,b$ in $V$ with $b\neq 0$. 

If $x$ and $y$ are two vectors in $V$ and $x \ne y$, then we denote by
$[x,y]$ the set  connecting $x$ and $y$. This is , $\{\alpha x + (1-\alpha )y\ | 0 \le \alpha \le 1\}$. One can easily check that $[x,y]$ is a closed line segment.

\subsubsection*{Remarks}
\begin{itemize}
\item An alternative, equivalent, definition is as follows: 
A (closed) line segment is a convex hull of two distinct points.
\item A line segment is connected, non-empty set.
\item If $V$ is a topological vector space, then a closed line segment
is a closed set in $V$. However, an open line segment is 
an open set in $V$ if and only if $V$ is one-dimensional. 
\item More generally than above, the concept of a line segment can be
  defined in an ordered geometry. 
\end{itemize}
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