Are these collections of subsets of a finite topological space path connected?

Question

Let the following be topologies of $X$ = {$a, b, c, d$}:

(i)$ τ_1$ = { $∅, X,$ {$a$}, {$c$}, {$a, b, d$}, {$b, c, d$}, {$b, d$}, {$a, c$} };

(ii) $τ_2$ = {$ ∅, X,$ {$a$}, {$a, d$}, {$a, b, c$} }.

It believe that (i) is not path connected because it is not connected, and that (ii) is path connected because it it connected. However I need to show that these are(not) path connected by showing there exists a(no) path between each points. I am struggling to identify these functions from $[0,1]$ to $X$.

Answer 1

Answer on (i):

Let $Y$ be a topological space and let $p:Y\rightarrow X$ be continuous function. Then the sets $p^{-1}\left(\left\{ b,d\right\} \right)$ and $p^{-1}\left(\left\{ a,c\right\} \right)$ are open and disjoint with $p^{-1}\left(\left\{ b,d\right\} \right)\cup p^{-1}\left(\left\{ a,c\right\} \right)=Y$.

If both sets are not empty then consequently $Y$ is not connected.

So no path in $X$ from $a$ to $b$ can exist, since interval $\left[0,1\right]$ is connected and a path from $a$ to $b$ is a continuous function $p:\left[0,1\right]\rightarrow X$ with $p(0)=a$ and $p(1)=b$.