Let $ varepsilon $ denotes the empty assembly.

Let $ X $ to be a topological space with topology $ T $.

Let $ langle B rangle $ denote 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 selecting $ F $, a set of cards of $ X $ at $ mathbb {R} $, declaring them arbitrarily continuous, then calculating the inverse images of the elements of $ T _ { mathbb {R}} $.

For example, I can get the standard topology on $ mathbb {R} ! times ! mathbb {R} $ by choosing the two functions $ f_1 $ and $ f_2 $ below

$$ f_1 (x, y) = x $$

$$ f_2 (x, y) = y $$

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

However, we cannot generate the SierpiĆski topology, $ { varepsilon, {0 }, {0, 1 } } $. In addition, the only finite topologies that we can generate by selecting functions in $ mathbb {R} $ are discrete topologies.

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

*What topologies can we generate by choosing a set of functions $ mathbb {R} $ to be continuous and then leveraging the existing topological structure of $ mathbb {R} $?*