line segment
Definition Suppose $V$ is a vector space over $\mathbb{R}$ or $\u2102$, and $L$ is a subset of $V$. Then $L$ is a line segment^{} if $L$ can be parametrized as
$$L=\{a+tb\mid t\in [0,1]\}$$ 
for some $a,b$ in $V$ with $b\ne 0$.
Sometimes one needs to distinguish between open and closed (http://planetmath.org/Closed) line segments. Then one defines a closed line segment as above, and an 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 $b\ne 0$.
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 , $\{\alpha x+(1\alpha )y0\le \alpha \le 1\}$. One can easily check that $[x,y]$ is a closed line segment.
Remarks

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An alternative, equivalent^{}, definition is as follows: A (closed) line segment is a convex hull of two distinct points.

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A line segment is connected, nonempty set.

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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 onedimensional.

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More generally than above, the concept of a line segment can be defined in an ordered geometry.
Title  line segment 
Canonical name  LineSegment 
Date of creation  20130322 14:19:01 
Last modified on  20130322 14:19:01 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  12 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 0300 
Classification  msc 5100 
Related topic  Interval 
Related topic  LinearManifold 
Related topic  LineInThePlane 
Related topic  CircularSegment 
Defines  open line segment 
Defines  closed line segment 