User:Anghel/Sandbox

Theorem
Let $A$ be a star convex subset of a normed 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 $A$.

Define $\mathbb I := \closedint 0 1$ as a closed real interval.

Let $\gamma : \mathbb I \to A$ be a loop in $A$ with base point $a$.

Let $\sigma : \mathbb I \to \set a$ be the constant function.

Constant Function is Continuous shows that $\sigma$ is continuous, so $\sigma$ is a loop with base point $a$.

Define $H: \mathbb I \times \mathbb I \to A$ by:


 * $\map H { s, t } = t \map {\gamma} s + \paren { 1-t } a$

By definition of star convex set, we have $\map H { s, t } \in A$ for all $\tuple { s, t } \in \mathbb I \times \mathbb I$.

Combination Theorem for Continuous Mappings shows that the mapping $H_{s'} : \mathbb I \to A $ defined by:


 * $ \map { H_{s'} } { t } = \map H { s' ,t }$ for $s' \in \mathbb I$ fixed

is continuous.

As $\gamma$ is continuous, it follows that the mapping $H_{t'} : \mathbb I \to A$ defined by:


 * $\map { H_{t'} } { s } = \map H { s, t_0 }$ for $t' \in \mathbb I$ fixed

is continuous.

Closed Real Interval is Compact shows that $\mathbb I$ is compact.

Heine-Cantor Theorem shows that all mappings $H_{s'}$ and $H_{t'}$ are uniformly continuous.

We check that $H$ is a path homotopy between $\gamma$ and $\sigma$:

It follows that all loops with base point $a$ are path-homotopic with $\sigma$.

Relative Homotopy is Equivalence Relation shows that all loops with base point $a$ are path-homotopic with each other.

This implies that the fundamental group $\map { \pi_1 } { A, a }$ is trivial.