# Topology/Path Connectedness

 Topology ← Connectedness Path Connectedness Compactness →

## 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).

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.

 Topology ← Connectedness Path Connectedness Compactness →