02_configuration_space

The space of all configurations is the configuration space or C-space.

topology

• topology space
• open set
• closed set
• interior point
• exterior point
• boundary point

$X=\mathbb{R}^n$ for any interger n is a topological space.

The intersection of open sets of $X$ is topology.

Hausdorff axiom: for any distinct $x_1,x_2\in X$, there exist open sets $O_,O_2$ such that $x_1\in O_1, x_2\in O_2$, and $O_1\bigcap O_2=0$. In other words, it is possible to separate x1 and x2 into nonoverlapping open sets.

Homeomorphism:[homɪə'mɔrfɪzəm],异质同胚.if $f \quad and\quad f^{-1}$ are continuous, then $f$ is called a homeomorphism. The spaces $X$ and $Y$ are said to be homeomorphic,denoted $X\cong Y$. interval homeomorphisms: Any open interval of R is homeomorphic to any other open interval.A subset $X\in \mathbb{R}$ and Y are homeomorphic if X can be mapped to Y via a nonsingular transformation.

Thus the topology of the robot does not change when it is translated or rotated.

manifold