Let $(X,\tau)$ be a topological space. If $A\subseteq X$ we define the following equivalence relation on $X$: $$\sim_A = \{(x,y) \in X^2: x=y \text{ or }\{x,y\}\subseteq A \}.$$

Let $(X,\tau)$ be an infinite space with the following property:

If $A\subseteq X$ and $|A|<|X|$ then $X\cong X/\sim_A$.

Does this imply that $(X,\tau)$ is one of the following?

- $(X,\tau)$ is discrete;
- $(X,\tau)$ is indiscrete;
- $X$ can be well-ordered with a relation $\leq$ such that $\tau$ consists of the upper sets of $(X,\leq)$.