A function that is continous but non constant between two particular topological spaces

Question

Find a non-constant function between $X,\tau_1$ and $(X,\tau)$ and $(X,\tau')$ where $\tau=\{X,12,34,\emptyset\}$ and $\tau'={X,123,12,1,\emptyset}$.

$f:(X,\tau)\to (X,\tau')$

I know that I need to find a function $f$ such that $f^{-1}(U)\in\tau$ for all $U\in \tau'$. However I am not seeing the non-constant function form.

Question:

What would you suggest as a function fulfilling the aforementioned conditions?

Thanks in advance!

Answer 1

$f(1)=1=f(2)$ and $f(3)=f(4)=2$ is not constant.

$f^{-1}[\emptyset]=\emptyset\in \tau$ (we can actually omit it, and also $f^{-1}[X]=X \in \tau$ as these hold for any function between two sets),

$f^{-1}[\{1\}]=\{1,2\} \in \tau$ $f^{-1}[\{1,2\}=f^{-1}[\{1,2,3\}]=X\in \tau$. So $f$ is continuous.

Answer 2

Exactly. In particular we need $f^{-1}(\{1\})\in\tau$, so it can be either $\{1,2\}$ or $\{3,4\}$ or $\emptyset$ (we exclude $X$ since $f$ ought not to be constant).
The first two options are symmetric, and you can continue this process (with either choice) to construct a continuous $f$..