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Lets say temporarily that a topological space $(X,\tau)$ is weird if there is a function $\varphi:X \to \tau$ such that for all $x \in X$:

  1. $x\in\varphi(x)$,
  2. $\{y\in X: x \in \varphi(y)\}$ 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?

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  • $\begingroup$ Very nice property! It's a bit like star-finite covering, no? (Possibly I mix up something...) $\endgroup$ – Dominic van der Zypen Nov 10 '17 at 5:58
  • $\begingroup$ Yes @DominicvanderZypen , this has the same flavor as a star-finite covering but quite stronger. Note that if $\varphi$ takes the constant value $X$ then it produces the star-finite covering $\{X\}$ but $\varphi$ does not have property 2 (unless $X$ is finite!). $\endgroup$ – Ramiro de la Vega Nov 10 '17 at 10:03

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