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On the wikipedia page about quotient spaces one can read the following:

In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of $X$ need not be inherited by $X/{\sim}$, and $X/{\sim}$ may have separation properties not shared by $X$.

This left me wondering if there are certain separation axioms for which the quotient is well-behaved, but I can seem to find any examples. I know that the quotient of a normal/regular space is not normal/regular. Does anyone have an example of an axiom that behaves well?

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Whether of not a quotient $X / \mathord{\sim}$ satisfies certain axioms of separation depends fairly little on the axioms of separation satisfied by $X$, but rather on properties of the equivalence relation $\sim$.

To take a very extreme example, consider the real line $\mathbb{R}$. As a metric space, it satisfies perhaps the strongest of the usual axiom of separation: perfect normality (it is normal (Hausdorff), and all of its closed subsets are Gδ). This property is hereditary (any subspace of a perfectly normal space is perfectly normal), and it sometimes denoted T6. However it is quite easy to take a quotient space of $\mathbb{R}$ which is not even T0.

Define $\sim$ on $\mathbb{R}$ by declaring $$x \sim y \Leftrightarrow \begin{cases} x, y \in \mathbb{Q},&\text{or} \\ x, y \notin \mathbb{Q}. \end{cases}$$ The quotient space $\mathbb{R}/\mathord{\sim}$ is then the two-point space $\{ Q,P \}$ with the trivial topology $\{ \varnothing , \{ Q,P \} \}$.

On the other hand, it is pretty easy to set up a quotient of a non-T0-space which is perfectly normal.

Define $X = \mathbb{R} \times \{ 0 , 1 \}$, and topologise it with the base consisting of all sets of the form $( a , b ) \times \{ 0 , 1 \}$ where $a < b$. Define $\sim$ on $X$ by $$\langle x,i \rangle \sim \langle y,j \rangle \Leftrightarrow x = y.$$ The resulting quotient space $X / \mathord{\sim}$ will be homeomorphic to the real line.

But there are connections that can be made between separation axioms holding in $X$, and properties of the equivalence relation $\sim$. For example:

  • $X / \mathord{\sim}$ is T1 iff every equivalence class of $\sim$ is a closed subset of $X$.
  • If $X$ is a normal space and $\sim$ is a closed equivalence relation on $X$,1 then $X / \mathord{\sim}$ is normal. (An equivalence relation $\sim$ on $X$ is closed if the natural mapping $X \to X / \mathord{\sim}$ is a closed mapping; equivalently, for every closed $F \subseteq X$ the set $\{ x \in X : ( \exists y \in F ) ( x \sim y ) \}$ is closed.)
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