I want to construct an open surjection from a $T_5$ space to a $T_1$ space which is not a $T_2$ space. A candidate I came up with is $\mathrm{denom}\colon \mathbb{Q} \to \mathbb{N}$ which sends a rational to its reduced denominator, the domain equipped with usual topology and the range with cofinite topology. This map is obviously surjective. It is also continuous because the inverse image of cofinite sets in $\mathbb{N}$ is $\mathbb{Q}$ minus a locally finite set of points.
To prove that it is open, we should prove that given a nonempty interval $I$, the set of $n$ such that there exists a rational in $I$ with reduced denominator $n$ is cofinite. An idea I have is that one may prove the largest gap between rationals with reduced denominator $n$ tends to $0$ with $n \to \infty$, but I'm not an expert in number theory so need some help.
The Jacobsthal function $j(n)$ is defined to be the smallest integer $m$ such that every set of $m$ consecutive integers contains at least one integer that is relatively prime to $n$. Your idea can be rephrased as the assertion that $j(n)/n$ tends to $0$ as $n$ tends to $\infty$.
This assertion is known to be true; indeed, Iwaniec proved that $j(n)/(\log n)^2$ is bounded above for all $n\ge2$, which is more than enough.
This result is a pretty heavy hammer to use (and is somewhat hard to pin down in the literature to boot), but I can't think of an argument that's too much simpler. It's pretty easy (once set up in the right way) to show via inclusion-exclusion that $j(n) \le 2^{\omega(n)}$ where $\omega(n)$ is the number of distinct prime factors of $n$; it's medium-hard (a graduate level exercise, but at least not a hard-to-locate research result) to show that $\omega(n) \le C(\log n)/\log(\log n)$ for some constant $C$. These two results together are enough to show that $j(n)/n\to 0$.