Let $X=\{p,q,r,s\}$ with topology induced by basis $\{p\}$,$\{r\}$,$\{p,q,r\}$, $\{p,r,s\}$. Then $X$ is not simply connected.
My Attempt:
$\{p\}$ is open set and $\{q\}$ is closed set.
Define $\phi:[0,1]\to X$, where
$$\phi(x)=\begin{cases} q & x=0,1,\\ p & x\in (0,1). \end{cases}.$$
$\phi$ is a continuous loop. But how to show it is not homotopic to constant?
Any Help will be appreciated.