# Topology/Path Connectedness

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## DefinitionEdit

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 $f(1)=x_1$

## ExampleEdit

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 ConnectednessEdit

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.

## ExercisesEdit

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 →