Topology/Path Connectedness

Definition

Template:TOC right 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 $f(t)=t\overrightarrow{a}+(1-t)\overrightarrow{b}$.
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 $f(t)=[(1-t)a_0+t{a}_1,(1-t)b_0+t{b}_1]$ 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

$f(x)=\{\begin{array}{cc}{f}_1(2x)& \text{if }x\in [0,\frac{1}{2}]\\ f_2(2x-1)& \text{if }x\in [\frac{1}{2},1]\end{array}$

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., $f:[0,1]\to X$ 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 $A=\{(x,f(x))|x\in ℝ\}\subset {ℝ}^2$, where $f(x)=\{\begin{array}{cc}0& \text{if }x\le 0\\ \sin (\frac{1}{x})& \text{if }x>0\end{array}$
   is connected but not path connected.

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