| Version 7 |
Version 6 |
| {\bf Definition} |
{\bf Definition} |
| Suppose $V$ is a vector space over $\sR$ or $\sC$, and $L$ is a subset of $V$. |
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 |
Then $L$ is a \emph{line segment} if $L$ can be parametrized |
| as |
as |
| $$L = \{ a+tb \mid t\in[0,1]\}$$ |
$$L = \{ a+tb \mid t\in[0,1]\}$$ |
| for some $a,b$ in $V$ with $b\neq 0$. |
for some $a,b$ in $V$ with $b\neq 0$. |
|
|
| Sometimes one needs to distinguish between open and closed |
Sometimes one needs to distinguish between open and closed |
| line segments. Then one defines a \emph{closed line segment} |
line segments. Then one defines a \emph{closed line segment} |
| as above, |
as above, |
| and an \emph{open line segment} as a subset $L$ that can be |
and an \emph{open line segment} as a subset $L$ that can be |
| parametrized as |
parametrized as |
| $$L = \{ a+tb \mid t\in(0,1)\}$$ |
$$L = \{ a+tb \mid t\in(0,1)\}$$ |
| for some $a,b$ in $V$ with $b\neq 0$. |
for some $a,b$ in $V$ with $b\neq 0$. |
|
|
| \subsubsection*{Remarks} |
\subsubsection*{Remarks} |
| \begin{itemize} |
\begin{itemize} |
| \item An alternative, equivalent, definition is as follows: |
\item An alternative, equivalent, definition is as follows: |
| A (closed) line segment is a convex hull of two distinct points. |
A (closed) line segment is a convex hull of two distinct points. |
| \item A line segment is connected, non-empty set. |
\item A line segment is connected, non-empty set. |
| \item If $V$ is a topological vector space, then a closed line segment |
\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 |
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. |
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 |
\item More generally than above, the concept of a line segment can be |
| defined in an ordered geometry. |
defined in an ordered geometry. |
| \end{itemize} |
\end{itemize} |
