---- > [!definition] Definition. ([[simply connected]]) > A [[topological space]] $X$ is called **simply connected** if is a [[path-connected]] space and if the [[fundamental group]] $\pi_{1}(X,x_{0})$ is trivial for some $x_{0} \in X$ (and [[change of basepoint isomorphism|hence for]] every $x_{0} \in X$). > [!equivalence] Equivalences. > - $X$ is [[simply connected]] iff any [[parameterized curve|loop]] in $X$ can be contracted to a point; > - $X$ is [[simply connected]] iff $X$ is [[path-connected]] and any two [[parameterized curve|paths]] in $X$ with the same initial and final points are [[path homotopy|path homotopic]]. > [!proof] Proof of Equivalences. > 1. (Immediate) Suppose [[fundamental group]] is trivial, then any [[parameterized curve|loop]] based at $x_{0}$ is [[path homotopy|path homotopic]] to the trivial loop $e_{x_{0}}$ since $\pi_{1}(X,x_{0})=([e_{x_{0}}])$. The [[path homotopy]] witnessing this is the desired 'contraction to a point'. Conversely, if any [[parameterized curve|loop]] based at $x_{0}$ is [[path homotopy|path homotopic]] to the trivial loop $e_{x_{0}}$, then all loops based at $x_{0}$ belong to $[e_{x_{0}}]$ rendering $\pi_{1}(X,x_{0})$ trivial. >2. [[in a simply connected space, any two paths having the same initial and final points are path homotopic|Here is one direction (easy)]]. Conversely suppose $X$ is [[path-connected]] and any two paths in $X$ with the same initial and final points are [[path homotopy|path homotopic]]. Then in particular any loop based at $x_{0}$ is path homotopic to $e_{x_{0}}$... hence $\pi_{1}(X,x_{0})=([e_{x_{0}}])$. > [!basicexample] > $\mathbb{R}^{n}$ is [[simply connected]]. Indeed, given any $x_{0} \in \mathbb{R}^{n}$, the [[straight-line homotopy]] witnesses that any [[parameterized curve|loop]] about $x_{0}$ is [[path homotopy|path homotopic]] to the constant path $e_{x_{0}}$. Hence $[e_{x_{0}}]$ consists of *all* loops about $x_{0}$, and so $\pi_{1}(X,x_0)=0$. > More generally, any convex subset of $\mathbb{R}^{n}$ (e.g., the unit ball) is [[simply connected]]. \ Even more generally, any [[star-convex set]] $(A,a_{0})$ in $\mathbb{R}^{n}$ is [[simply connected]]. For it is [[path-connected]] (connect any two points with the 'broken line' passing through $a_{0}$) and the [[fundamental group]] $\pi_{1}(A,a_{0})$ is trivial, as witnessed by the following analog to the [[straight-line homotopy]]: given a [[parameterized curve|loop]] $f$ about $a_{0}$, define $\begin{align} F: I \times I & \to A \subset \mathbb{R}^{n} \\ F(s,t) & := (1-t)f(s) + ta_{0}. \end{align}$ > Then $F$ is a [[path homotopy]] between $f$ and $e_{a_{0}}$; indeed, we verify >- $F(s, 0)=f(s)$; >- $F(s,1)=a_{0}=e_{a_{0}}(s)$; >- $F(0,t)=(1-t)\overbrace{f(0)}^{=a_{0}}+ta_{0}=a_{0}-ta_{0}+ta_{0}=a_{0}$; >- $F$ is [[continuous]] on $I \times I$ because$F(s,t)=F_{1}(s,t)F_{2}(s,t)+a_{0}F_{3}(s,t)$, where $\begin{align} F_{1}(s,t):= & 1-t \\ F_{2}(s,t):= & f(s) \\ F_{3}(s,t):= & t \end{align}$ and therefore $F$ equals the sum of products of [[continuous]] functions. > >Since $f$ was arbitrary, we conclude that any loop about $a_{0}$ is [[path homotopy|path homotopic]] to $e_{a_{0}}$, hence $\pi_{1}(A, a_{0})=[e_{a_{0}}]=0$. ^acd82f > [!basicexample] > The set $X=\{ 0,1 \}$ equipped with the [[topological space|topology]] $\tau=\{ \emptyset, \{ 0 \}, X \}$ is [[simply connected]]. Clearly $X$ is [[path-connected]]: constant [[parameterized curve|paths]] connect $0$ to $0$ and $1$ to $1$, and e.g. the path $\begin{align} \gamma : [0,1] & \to X \\ t & \mapsto \begin{cases} 0 & t \in \left[ 0, \frac{1}{2} \right) \\ 1 & t \in \left[ \frac{1}{2}, 1 \right] \end{cases} \end{align}$ connects $0$ to $1$. ($\gamma$ is [[continuous]] because >- $\gamma ^{-1}(\{ 0 \})=[0, \frac{1}{2}) = (-\frac{1}{2}, \frac{1}{2}) \cap [0,1]$ is [[subspace topology|open in]] $[0,1]$ >- $\gamma ^{-1}(\emptyset)=\emptyset$ and $\gamma ^{-1}(X)=[0,1]$ are trivially open in $[0,1]$ and clearly $\gamma(0)=0$ and $\gamma(1)=1$. So $\gamma$ is indeed a [[parameterized curve]] from $0$ to $1$.) We just have to show that any two paths in $X$ from $0$ to $1$ are [[path homotopy|path homotopic]]. So let $f,g: I \to X$ be two paths in $X$ with $f(0)=0=g(0)$ and $f(1)=1=g(1)$. Define $\begin{align} & F: I \times I \to X \\ & F(s,t) := \begin{cases} f(s) & t=0 \\ 0 & s \neq 1 \text{ and } t \in (0,1) \\ 1 & s=1 \text{ and } t \in (0,1) \\ g(s) & t=1. \end{cases} \end{align}$ To show $F$ is [[continuous]], it suffices to verify that $F^{-1}(\{ 0 \})$ (shaded blue) is open in $I \times I \subset \mathbb{R}^{2}$. ![[CleanShot 2024-03-24 at 20.44.21.jpg|400]] We will show that $F^{-1}(\{ 0 \})$ equals its own [[topological interior|interior]]. > To show that $F^{-1}(\{ 0 \}) \subset \text{int }F^{-1}(\{ 0 \})$: Let $p \in F^{-1}(\{ 0 \})$ and proceed by cases. Since all of $\text{int}(I \times I)$ is contained in $F^{-1}(\{ 0 \})$, if $p \in (0,1) \times (0,1)$ we are immediately done. Otherwise $p \in \text{Bd}(I \times I)$. Suppose $p=(p_{s}, p_{t}) \in I \times \{ 1 \}$. Since $f$ is [[continuous]], $p \in (a,b) \times \{ 1 \}$ for some open interval $(a,b) \subset [0,1]$. Now $U=(a,b) \times (\frac{1}{2}, \frac{3}{2})$ is an open subset of $\mathbb{R}^{2}$, hence $O=U \cap (I \times I)$ is open in $I \times I$. Since $p \in O$ and $O \subset F^{-1}(\{ 0 \})$ we conclude that $p \in \text{int } F^{-1}(\{ 0 \})$. A symmetric argument goes through for $p \in I \times \{ 0 \}$. The final case is $p \in \{ 0 \} \times I$. Since all of $\{ 0 \} \times I$ belongs to $F^{-1}(\{ 0 \})$, we can put a open (in $\mathbb{R}^{2}$) ball $B$ of radius $\min(p_{t}, 1-p_{t})$ about $p$; then $B \cap (I \times I)$ is an open set in $F^{-1}(\{ 0 \})$ containing $p$. So $p \in \text{int }F^{-1}(\{ 0 \})$. > That $\text{int }F^{-1}(\{ 0 \}) \subset F^{-1}(\{ 0 \})$ is immediate from the definition of [[topological interior|interior]]. > To show that $F$ is a [[path homotopy]] between $f$ and $g$, we verify >- $F(s, 0)=f(s)$ >- $F(s,1)=g(s)$ >- $F(0, t)=0=f(0)=g(0)$ >- $F(1,t)=1=f(1)=g(1)$. ^a78e82 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```