I'm new to equivalence classes here, so I hope someone could help me out
Mathworld (http://mathworld.wolfram.com/EquivalenceClass.html) defined equivalence classes as "a subset of the form$ (x \in X:x\sim a),$ where a is an element of X and the notation $"x\sim y"$ is used to mean that there is an equivalence relation between $x$ and $y$.
I wonder if that is the case, then is the equivalence classes of the relation $x\sim y$ if $x=y$ in the real is just the set of $x$ itself? Then would the quotient space $R / \sim$ be an empty set?
If the equivalence relation is the equals relation on $\mathbb{R}$, then the equivalence class of $x \in \mathbb{R}$ is $\{x\}$, the set containing only $x$.
The quotient space $\mathbb{R}/\mathord{=}$ is bijective to $\mathbb{R}$ by the mapping $x \leftrightarrow \{x\}$.