Is the following subset connected? Is the following subset connected?
$A=\{a,d,e\}$ subset of a topological space $(X,\tau)$, $X=\{a,b,c,d,e\}$ and $\tau = \{X,\phi,\{a,b,c\},\{c,d,e\},\{c\}\}$.

My ans:

The closed sets of $\tau$ are given by $\{\phi,X,\{d,e\},\{a,b\},\{a,b,d,e\}\}$. Clearly the set $A \neq U \cup V$ for some U and V, both open (closed) in $\tau$.
Hence, the set $A$ is connected.

But my solution key says this set is disconnected. what am I doing wrong here and also, is my idea to solve this question correct? Thanks.
You have that
$$A =\left(\lbrace a,b,c\rbrace \cap A\right) \cup \left(\lbrace c,d,e \rbrace) \cap A\right)$$
so $A$ can be written as the union of two open disjoints subsets of itself, for the topology on $A$ induced by the given topology on $X$. Therefore $A$ is not connected.
$A\subseteq\{a,b,c\}\cup\{d,e,c\}= U\cup V$ provides a disconnection of $A$. Although $U\cap V\ne\emptyset$,  this is not required for a disconnection. What is required is that $U\cap\{d,e\}=\emptyset=V\cap\{a\}$.