Let $X$ be a set and $\tau_1,\tau_2$ two topologies on $X$ such that every open set of $\tau_1$ is also open in $\tau_2$. Prove that if $(X,\tau_2)$ is connected then $(X,\tau_1)$ is also connected. Prove this also holds for path connectedness.
Suppose $(X,\tau_1)$ is not connected. Then there are two disjoint nonempty subsets $\theta_1,\theta_2 \in \tau_1$ such that $X=\theta_1 \cup \theta_2$. Every open subset in $\tau_1$ is in $\tau_2$, so we have two disjoint subsets $\theta_1,\theta_2 \in \tau_2$ such that $X=\theta_1 \cup \theta_2$. This contradicts the connectedness of $(X,\tau_2)$.
I'm struggling to prove the result for path connected sets. My definition for path connected sets is:
$(X,\tau)$ is path connected if for any two points $x,y \in X$ there exists a continuous function $f:[0,1] \rightarrow X$ such that $f(0)=x, f(1)=y$.
Path connectedness is not defined in terms of open sets so I don't know how to use the definition in this problem.