Let $X=[-1,1]\subseteq \mathbb{R}$ endowed with the topology
$\tau=\{U\subseteq X \quad|\quad 0\notin U\} \quad \cup \quad \{U\subseteq X \quad | \quad (-1,1)\subseteq U\}$.
Prove or disprove that $(-1,1)$ is a path connected subspace of $X$.
I've proved that $(-1,1)$ is connected showing that there are no open, non empty, disjointed sets in $X$ such that $(-1,1)$ is the union of them.
For the path connection, I try with the following function
$\gamma: [0,1] \to (-1,1), \quad \gamma(t)=ty+(1-t)x$
but I think that it doesn't work because I can't show that is continuous.
Anyone who can give me suggestions? Thanks.