Paths give explicit continuous motion inside a metric space. We now compare path-connectedness and ordinary connectedness. The main theorem says that every path-connected metric space is connected. The converse is false, and this distinction is one of the most important lessons in connectedness theory. Path-connectedness and ordinary connectedness coincide in many familiar regions, but not in all metric spaces. The focus keyword path-connectedness and ordinary connectedness helps students remember both the implication and the counterexample. :::theorem[Path-Connected Spaces Are Connected] Let $(X,d)$ be a metric space. If $X$ is path-connected, then $X$ is connected. ::: :::proof Given that $(X,d)$ is path-connected. To prove that $X$ is connected. Choose a point $x_0\in X$. For each $x\in X$, there exists a path $f_x:[0,1]\to X$ such that $f_x(0)=x_0$ and $f_x(1)=x$. Since $[0,1]$ is connected and $f_x$ is continuous, the image $f_x([0,1])$ is connected. Each such image contains the common point $x_0$. Therefore $\bigcup_{x\in X}f_x([0,1])$ is connected as a union of connected sets with a common point. But every point $x\in X$ belongs to $f_x([0,1])$. Hence $X=\bigcup_{x\in X}f_x([0,1])$. Therefore $X$ is connected. $\square$ ::: :::remark Every path-connected space is connected. However, a connected space need not be path-connected. The set $\{(0,y):-1\leq y\leq 1\}\cup\{(x,y):y=\sin(1/x),\ 0<x\leq 1\}$ is connected but not path-connected. ::: :::solved-problem Let $X=C[0,1]$ and define $d(x,y)=\sup\{|x(t)-y(t)|:0\leq t\leq 1\}$. Show that $(C[0,1],d)$ is connected. ::: :::solution Given that $X=C[0,1]$ with the supremum metric. To prove that $X$ is connected. It is enough to prove that $X$ is path-connected. Let $x,y\in C[0,1]$. Define $\gamma:[0,1]\to C[0,1]$ by $\gamma(s)=sx+(1-s)y$. This means that for each $t\in[0,1]$, $\gamma(s)(t)=sx(t)+(1-s)y(t)$. Since $x$ and $y$ are continuous, $\gamma(s)$ is continuous for each $s\in[0,1]$. Thus $\gamma(s)\in C[0,1]$. Also $\gamma(0)=y$ and $\gamma(1)=x$. Let $s,r\in[0,1]$. Then $$\begin{align*} \gamma(s)-\gamma(r)&=(s-r)(x-y),\\ d(\gamma(s),\gamma(r))&=|s-r|d(x,y). \end{align*}$$ This shows that $\gamma$ is continuous. Hence any two elements of $C[0,1]$ can be joined by a path. Therefore $C[0,1]$ is path-connected. Since every path-connected metric space is connected, $(C[0,1],d)$ is connected. $\square$ ::: :::example[Convex Sets Are Connected] Every convex subset $C$ of a normed linear space is path-connected by the straight-line path $(1-t)x+ty$. Since every path-connected metric space is connected, every convex subset of a normed linear space is connected. $\square$ ::: :::solved-problem Show that the unit interval $[0,1]$ is path-connected and hence connected. ::: :::solution Given that $[0,1]\subseteq\mathbb{R}$. To prove that $[0,1]$ is path-connected and hence connected. Let $a,b\in[0,1]$. Define $f:[0,1]\to[0,1]$ by $f(t)=(1-t)a+tb$. Since $[0,1]$ is convex, $f(t)\in[0,1]$ for every $t\in[0,1]$. The map $f$ is continuous, $f(0)=a$, and $f(1)=b$. Therefore $[0,1]$ is path-connected. Hence $[0,1]$ is connected. $\square$ ::: :::exercise [1] Prove that every path-connected metric space is connected. [2] Give an example of a connected space that is not path-connected. [3] Show that $C[0,1]$ with the supremum metric is path-connected. [4] Explain why the straight-line path is useful in convex sets. ::: :::answer [1] Express the space as a union of path images from one fixed point. [2] The closure of the oscillating sine curve is connected but not path-connected. [3] Use $\gamma(s)=sx+(1-s)y$. [4] Convexity ensures that the whole segment between two points stays in the set. ::: :::faq[Frequently Asked Questions] Q: Does connectedness imply path-connectedness? A: Not in general. Q: Why does the proof use a fixed basepoint? A: It makes the whole space a union of connected path images sharing one common point. Q: Why is $C[0,1]$ connected? A: It is path-connected through straight-line interpolation of functions. ::: :::call-to-action[Continue to Path-Connected Images and Unions] subtitle: After path-connectedness and ordinary connectedness, study how path-connectedness behaves under maps and unions. button: Next Lesson | /functional-analysis/metric-spaces/connectedness-and-path-structure-in-metric-spaces/path-connected-images-and-unions button-ghost: Previous Lesson | /functional-analysis/metric-spaces/connectedness-and-path-structure-in-metric-spaces/paths-and-path-connected-sets :::