12 - Path Connected and Connected

Path Connected and Connected

A path-connected metric space is connected

Proof

Suppose that $X$ is path-connected

Let $f:X\to \{0,1\}$
Let $a,b\in X$ so as $X$ is path connected there exists path $\gamma :[0,1]\to X$ such that

$$\gamma (0)=a,\gamma (1)=b$$

Consider composition $f\circ \gamma$ which is a continuous function from $[0,1]$ to $\{0,1\}$
Hence as $[0,1]$ is connected by Connectedness of Intervals then $f\circ \gamma$ is constant
Thus

$$f(a)=(f\circ \gamma )(0)=(f\circ \gamma )(1)=f(b)$$

As $a$ and $b$ are arbitrary then $f$ is constant
Hence by Equivalent statements for connected metric spaces then $X$ is connected