Problem: Let $X$ be any topological space. Let $A\subseteq X$ be nonempty. For $x,y\in A$, define $x\sim y$ if and only if there is a connected subset $C\subseteq A$ such that $x,y\in C$. For $x\in A$, define \[ C(x) \coloneqq \left\{ y\in A: y\sim x \right\}. \] Then which of the following is true.
Solution: Note that here $C(x)$ is the equivalence class of $x$, which contains all the elements which are related to $y$. We have the following property for the equivalence classes. \begin{equation}\label{eq:20Apr2023-1} C(x) \cap C(y) \neq \emptyset \iff C(x) = C(y) \iff x\sim y. \end{equation} Let us prove this property. We have $z\in C(x) \cap C(y) $ if and only if $z\sim x$ and $z\sim y$. By the transitive property we have $x\sim y$ this implies $C(x) = C(y)$. Hence we proved the claim.
By using the equation \eqref{eq:20Apr2023-1}, we conclude that options $(B),(C)$ and $(D)$ are correct. Note that option $(A)$ need not be true. For example, let $X= \mathbb{R} $ with the euclidean topology and $A=[-1,1]$. Then note that $C(-0.5) = C(0.5)$ as we can take the $C= (-0.6,0.6)$, but $x \neq y$.