Measurability and continuity for general topological spaces

Question

Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ saturated if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K \text{ is compact and saturated}\}$.

Let $(X_i, \tau_i)$ be topological spaces for $i=1,2$. A function $f:X_1\to X_2$ is said to be measurable if $S\in \sigma(X_2,\tau_2)$ implies $f^{-1}(S)\in\sigma(X_1,\tau_1)$.

Are there topological spaces $X,Y$ and a continuous function $f: X\to Y$ such that $f$ is not measurable in the above sense?

Answer 1

I'm assuming that by "compact" you only mean which satisfies the finite cover properties, and not compact and Hausdorff. in this case the following produces a counterexample:

Let $X_1$ be $\mathbb{R}$ with the ordinary topology, (and assume that there is at least one non Borel subset of $\mathbb{R}$...).

Let $X_2$ be $\mathbb{R}$ with the co-finite topology. then the identity map from $X_1$ to $X_2$ is continuous, but any subset of $X_2$ is saturated and compact hence in the tribe you define.