Lets say temporarily that a topological space $(X,\tau)$ is weird if there is a function $\varphi:X \to \tau$ such that for all $x \in X$:
- $x\in\varphi(x)$,
- $\{y\in X: x \in \varphi(y)\}$ is finite.
A function satisfying the first condition is sometimes called a neighborhood assignment. Note that discrete spaces are weird and indiscrete spaces are not. Any countable Hausdorff space is weird but $\omega_1$ with the order topology is not.
Have weird spaces been studied before? how are they called? Is there a standard name for a function satisfying both conditions?
