Identification of a Subset to a Point

Question

If $A$ is a subspace of a topological space $S$, we can define a relation $∼$ on S by declaring $$x ∼ x\quad\text{for all}\quad x\in S$$ (so the relation is reflexive) and $$x ∼ y\quad\text{for all}\quad x, y\in A.$$

Question 1. On the book it says that this is an equivalence relation on $S$. Why?

Question 2. Who are the equivalence classes?

Thanks!

Answer 1

Check the definitions to see it is an equivalence relation:

• reflexivity is given.
• If $x \sim y$ then either we're in the case $x \sim x$ (option 1) and so $y=x$ and $y \sim x$ is again given. Or we're in the case $x,y \in A$ and so then also $y \sim x$. The statement $x\in A \land y \in A$ is symmetric in $x$ and $y$.
• For proofs of transitivity we can always assume WLOG that all three points involved are different: in this case $x \sim y$ and $y\sim z$ then implies $x,y\in A$ and $y,z\in A$, so clearly also $x,z\in A$ (we jus state less info) so $x \sim z$.

The equivalence classes of course are $A$ and all $\{x\}$ where $x \notin A$. So $A$ becomes a new point in the quotient space and all other are left untouched.