Are there any half-homogeneous topological spaces with dense orbits?

Question

A topological space $X$ is half-homogeneous ($\frac{1}{2}$-homogeneous) if the action of the homeomorphism group of $X$ splits it into two different orbits. Wandering in my mind, it ocurred to me that the orbit space $\frac{X}{O}$, which is the quotient where $O$ is the partition into orbits, has two points, and so can have at most three quotient topologies.

It is a discrete space whenever both orbits in $X$ are closed, hence clopen. (for example, the disjoint union of $\mathcal{S}^1$ and an open ball in $\mathbb{R}^2$).

It is a Sierpinski space whenever one of the orbits is closed but not open (the case of $[0,1)$).

And it is an indiscrete space whenever none of the orbits are closed. This is a strange one.

If both orbits in $X$ are not closed, then both must be dense, and so their interiors must be empty. But I cannot think of a space with this property... Does there even exist one?

Answer 1

A simple example is an infinite space $X$ whose topology consists of all cofinite subsets that contain a fixed point $x ∈ X$.

I think another example is a regular dendrite with dense set of branch-points and with endpoints removed, so the two orbits are the set of all brachpoints and the set of all ordinary points.