Topology/Path Connectedness

Definition

A topological space $X$ is said to be path connected if for any two points $x_{0},x_{1}\in X$ there exists a continuous function $f:[0,1]\to X$ such that $f(0)=x_{0}$ and $f(1)=x_{1}$

Example

1. All convex sets in a vector space are connected because one could just use the segment connecting them, which is $f(t)=t{\vec {a}}+(1-t){\vec {b}}$.
2. The unit square defined by the vertices $[0,0],[1,0],[0,1],[1,1]$ is path connected. Given two points $(a_{0},b_{0}),(a_{1},b_{1})\in [0,1]\times [0,1]$ the points are connected by the function $f(t)=[(1-t)a_{0}+ta_{1},(1-t)b_{0}+tb_{1}]$ for $t\in [0,1]$.
   The preceding example works in any convex space (it is in fact almost the definition of a convex space).

Adjoining Paths

Let $X$ be a topological space and let $a,b,c\in X$. Consider two continuous functions $f_{1},f_{2}:[0,1]\to X$ such that $f_{1}(0)=a$, $f_{1}(1)=b=f_{2}(0)$ and $f_{2}(1)=c$. Then the function defined by

$f(x)=\left\{{\begin{array}{ll}f_{1}(2x)&{\text{if }}x\in [0,{\frac {1}{2}}]\\f_{2}(2x-1)&{\text{if }}x\in [{\frac {1}{2}},1]\\\end{array}}\right.$

Is a continuous path from $a$ to $c$. Thus, a path from $a$ to $b$ and a path from $b$ to $c$ can be adjoined together to form a path from $a$ to $c$.

Relation to Connectedness

Each path connected space $X$ is also connected. This can be seen as follows:

Assume that $X$ is not connected. Then $X$ is the disjoint union of two open sets $A$ and $B$. Let $a\in A$ and $b\in B$. Then there is a path $f$ from $a$ to $b$, i.e., $f:[0,1]\rightarrow X$ is a continuous function with $f(0)=a$ and $f(1)=b$. But then $f^{-1}(A)$ and $f^{-1}(B)$ are disjoint open sets in $[0,1]$, covering the unit interval. This contradicts the fact that the unit interval is connected.

Exercises

1. Prove that the set $A=\{(x,f(x))|x\in \mathbb {R} \}\subset \mathbb {R} ^{2}$, where $f(x)=\left\{{\begin{array}{ll}0&{\text{if }}x\leq 0\\\sin({\frac {1}{x}})&{\text{if }}x>0\\\end{array}}\right.$
   is connected but not path connected.