Let $A \subset X$. We can define an equivalence relation on $X$ by $x_1 \sim x_2$ if $x_1 = x_2$ or if both $x_1$ and $x_2$ are in $A.$ Then we can naturally define the quotient topology $X/A$ on the set of all the equivalence classes generated by the equivalence relation $\sim$. My question is this: if $A = \emptyset$, then why is $\emptyset$ also an equivalence class? By definition, equivalence classes are generated by a point in the set $X$, i.e., for some $a \in X$, we have $[a]:= \{x \in X \ | \ a \sim x\}$. But what element generates the empty set?
I would appreciate any help. Thank you.
Best Answer
It depends on how you define quotient. Quotient are also defined categorically as adjunction spaces. $X/A$ is the pushout of two maps from $A$. The first map is the inclusion $A \hookrightarrow X$. The second map is the unique map to a point $A \rightarrow p$. With this definition, we get the counter-intuitive result that $A/\emptyset$ is $A$ together with an additional point. If you take the platonic view of mathematical objects, then this may simply be an artefact of this definition. A description of adjunctions spaces can be found here: https://math.jhu.edu/~jmb/note/pushout.pdf