Let $\varepsilon$ denote the empty set.

Let $X$ be a topological space. We will be considering multiple possible topologies on $X$.

Let $\langle B \rangle$ denote the closure under arbitrary unions, including unions of zero elements, and finite intersections of a set of sets, $B$.

Let $T\_{\mathbb{R}}$ denote the standard topology on the reals.

Which topologies can be generated by picking $F$, a set of maps from $X$ to $\mathbb{R}$, arbitrarily declaring them to be continuous, and then computing the inverse images of elements of $T\_{\mathbb{R}}$.

For instance, I can get the standard topology on $\mathbb{R}\!\times\!\mathbb{R}$ by picking the two functions $f\_1$ and $f\_2$ below

$$ f\_1(x, y) = x $$
$$ f\_2(x, y) = y $$

Computing $\langle f^{-1}\_1(T\_{\mathbb{R}}), f^{-1}\_2(T\_{\mathbb{R}}) \rangle$ gives me the topology I want. I can construct any epsilon ball in $\mathbb{R}\!\times\!\mathbb{R}$ that I want by unioning together a countable set of open squares. The epsilon balls form a basis of the standard topology on $\mathbb{R}\!\times\!\mathbb{R}$, so I'm done.

However, we can't generate the Sierpiński topology, $\{ \varepsilon, \{0\}, \{0, 1\}\}$.

Any potential function $f$ must send $0$ to a single real number and $1$ to a single real number and every real number is in some open set.

*Which topologies can we generate by picking some set of functions to $\mathbb{R}$ to be continuous and then leveraging the existing topological structure of $\mathbb{R}$?*