Connected Sets in Rn

Subjects: Vector Analysis
Links: Perfect and Connected Sets in R, Topological Connectedness

Def: Let $A\subseteq {\mathbb{R}}^n$. We say that $A$ is disconnected if there exist $B,C\subseteq {\mathbb{R}}^n$, nonempty such that:

• $A=B\cup C$
• $\mathrm{cl}(B)\cap C=B\cap \mathrm{cl}(C)=\varnothing$

A set is connected if it not disconnected

Def: Let $x,y\in {\mathbb{R}}^n$. We define the straight line segment that joins $x$ with $y$, denoted with $[x,y]$ and defined by

Th: Let $B,C\subseteq {\mathbb{R}}^n$ such that $x\in B$ and $y\in C$. If $B$ and $C$ are separated, then $[x,y]⊈B\cup C$.

Def: Let $A\subseteq {\mathbb{R}}^n$. We say that $A$ is convex if for every pair $x,y\in A$, then $[x,y]\subseteq A$

Th: Let $A\subseteq {\mathbb{R}}^n$. If $A$ is convex, then $A$ is connected

Def: Let $x=x_0,x_1,x_2,\dots ,x_k=y\in {\mathbb{R}}^n$, and we will that the set $\bigcup _{i=1}^n[x_i,x_{i-1}]$ is a polygonal curve that connects $x$ and $y$.

Def: Let $A\subseteq {\mathbb{R}}^n$. $A$ is polygonally connected if for every pair of points $x,y\in A$ exists a polygonal curve that connects $x$ and $y$ and is contained in $A$.

Th: Let $A\subseteq {\mathbb{R}}^n$ be a nonempty open set. $A$ is connected, iff$A$ is polygonally connected.

Prop: Let $A\subseteq {\mathbb{R}}^n$. $A$ is disconnected iff there’s $U,V\subseteq {\mathbb{R}}^n$ open sets such that:

• $A\subseteq U\cup V$
• $A\cap U\ne \varnothing$ and $A\cap V\ne \varnothing$
• $A\cap U\cap V=\varnothing$

Cor: Let $A\subseteq {\mathbb{R}}^n$. $A$ is connected iff there’s no $U,V\subseteq {\mathbb{R}}^n$ open sets such that:

• $A\subseteq U\cup V$
• $A\cap U\ne \varnothing$ and $A\cap V\ne \varnothing$
• $A\cap U\cap V=\varnothing$

Path-Connected Sets

Def:************ A set $A\subseteq {\mathbb{R}}^n$ is path-connected if for every pair of points $x,y\in A$ there exists a continuous fucntion $\gamma :[0,1]\to {\mathbb{R}}^n$ with $\gamma (0)=1$ and $\gamma (1)=b$. We call $\gamma$ a ****path joining $a$ and $b$.

Prop:************ A path-connected set is connected.

Th: Let $A\subseteq {\mathbb{R}}^n$ an nonempty open set. $A$ is connected iff $A$ is path-connected.

Th: Let $A\subseteq {\mathbb{R}}^n$ be an open connected set, and $x,y\in A$, then there’s a differentiable path $\gamma :[0,1]\to {\mathbb{R}}^n$ with $\gamma (0)=x$ and $\gamma (1)=y$

Path-Covering Lemma

Suppose $\gamma :[a,b]\to U$ is a continuous path from the interval into an open subset $U$ of ${\mathbb{R}}^n$. Then there are a number $\rho >0$ and a partition $P$ of the interval $[a,b]$ such that:

• $B_{\rho }(t_k)\subseteq U$ for $k\in \{1,\dots ,m\}$
• $\gamma (t)\in B_{\rho }(\gamma (t_0))$ for $t_0\le t\le t_1$
• $\gamma (t)\in B_{\rho }(\gamma (t_k))$ for $t_{k-1}\le t\le t_{k+1}$
• $\gamma (t)\in B_{\rho }(\gamma (t_m))$ for $t_{m-1}\le t\le t_m$