There is a very wide series of questions I have been thinking about and I am wondering if there is any literature on this type of structures.

Let's start with the set of all functions $\mathcal{F}: \mathbb{R} \to \mathbb{R}$. The subsets of this set form a lattice (based on set inclusion). Let $\mathbb{P}_{n}$, $\mathcal{A}$, and $\mathcal{C}$ be the rings of polynomials with degree at most $n$, analytic functions, and continuous functions respectively. So that in the lattice structure we have $\mathbb{P}_0 \subset \mathbb{P}_3 \subset \mathcal{A} \subset \mathcal{C} \subset \mathcal{F}$. For simplicity limit the elements of this lattice to those sets which form a ring of functions. I am not sure why, but this seems like a good restriction.

Suppose we are given a subset of $\mathcal{F}$ which is a ring, let's call this $\mathcal{R}$. Suppose we also a set of points $\mathbb{S} \subset \mathbb{R}$. Then we say $\mathbb{S}$ is a *separating* set of $\mathcal{R}$, if for any $f,g \subset \mathcal{R}$ there is an $x \in \mathbb{S}$, such that $f(x) \neq g(x)$. Essentially the values of a function on $\mathbb{S}$ uniquely determine the function in $\mathcal{R}$.

So, let's look at what some of these sets look like.

Ring of Functions | Separating set |
---|---|

$\mathbb{P}_0$ | $\{x_0 | x_0 \in \mathbb{R}\}$, any singleton |

$\mathbb{P}_n$ | $\{x_0, x_1, ..., x_n | x_i \in \mathbb{R}, x_i \neq x_j \text{ if } i \neq j \}$, any set of $n$ points |

$\mathcal{A}$ | $\mathbb{S} \subset \mathbb{R}$, such that $\mathbb{S}$ has a limit point. |

$\mathcal{C}$ | $\mathbb{S} \subset \mathbb{R}$, such that $\mathbb{S}$ is dense in $\mathbb{R}$ |

$\mathcal{F}$ | $\mathbb{R}$ |

These sets seem to be capturing some kind of topological property. I am not sure about what kind of topological equivalence would capture these sets. On the other hand, these sets definitely form a lattice as well through inclusion. There is an anti-correspondence between these sets and their corresponding function rings.

Do these functions sets have to be rings? No, but it seems like a good idea.

Does the base field have to be $\mathbb{R}$? No, but it is a good starting place.

What I am interested in is if the topological property can be well defined and some kind of lattice be determined on it from which we can recover the lattice of functions. It will probably dictate the structure of the function sets (rings, groups, etc).

It seems the topological equivalence I am looking for is just a homeomorphism. Dense sets will remain dense, sets with limit points preserve those. Am I thinking correctly?

What do we know about the lattice of subsets of $\mathbb{R}$ upto homeomorphic equivalence?