User:Anghel/Sandbox

Theorem
Let $A$ be a star convex subset of a vector space $V$ over $\R$ or $\C$.

Then $A$ is simply connected.

Proof
Star Convex Set is Path-Connected shows that $A$ is path-connected.

Let $a \in A$ be a star center of $A$.

Let $\gamma_1, \gamma_2 : \closedint 0 1 \to A$ be two loops in $A$ with base point $a$.

Define $H: \closedint 0 1 \times \closedint 0 1 \to A$ by:


 * $\map H { s, t } = \leftset { \begin{array}{ll} 2 t a - \paren { 1 - 2 t } \map { \gamma_1 }{ t } & \mathrm{for} \: t \in \closedint 0 { \dfrac 1 2 } \\  \end{array} }$

$$

Fundamental Group is Independent of Base Point for Path-Connected Space