An open question on MSE, http://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a topology over that field such that the continuous functions from the field to itself are precisely the polynomials (and whether such a topology exists for $\Bbb Q$, $\Bbb R$, or $\Bbb C$). Another question, linked from there, asks the same about holomorphic functions. The common notion is this:

Given a set $X$ and a set $\mathscr C$ of functions from $X$ to $X$, what are necessary and/or sufficient conditions under which there is a topology $\mathscr T$ on $X$ such that a function $f\colon X\to X$ is $\mathscr T$-$\mathscr T$ continuous iff $f\in \mathscr C$.

It is immediately clear that $\mathscr C$ must contain the identity function and all constant functions, and must be closed under composition. Nothing else seems terribly obvious. What is known about this?

Edit: for the purpose of answering the specific questions on MSE, even results limited to connected, bihomogeneous $T_1$ spaces would be very helpful.

An open question on MSE, https://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a topology over that field such that the continuous functions from the field to itself are precisely the polynomials (and whether such a topology exists for $\Bbb Q$, $\Bbb R$, or $\Bbb C$). Another question, linked from there, asks the same about holomorphic functions. The common notion is this:

Given a set $X$ and a set $\mathscr C$ of functions from $X$ to $X$, what are necessary and/or sufficient conditions under which there is a topology $\mathscr T$ on $X$ such that a function $f\colon X\to X$ is $\mathscr T$-$\mathscr T$ continuous iff $f\in \mathscr C$.

It is immediately clear that $\mathscr C$ must contain the identity function and all constant functions, and must be closed under composition. Nothing else seems terribly obvious. What is known about this?

Edit: for the purpose of answering the specific questions on MSE, even results limited to connected, bihomogeneous $T_1$ spaces would be very helpful.

An open question on MSE, http://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a topology over that field such that the continuous functions from the field to itself are precisely the polynomials (and whether such a topology exists for $\Bbb Q$, $\Bbb R$, or $\Bbb C$). Another question, linked from there, asks the same about holomorphic functions. The common notion is this:

Given a set $X$ and a set $\mathscr C$ of functions from $X$ to $X$, what are necessary and/or sufficient conditions under which there is a topology $\mathscr T$ on $X$ such that a function $F\colon X\to X$ is $\mathscr T$-$\mathscr T$ continuous iff $f\in \mathscr C$.

It is immediately clear that $\mathscr C$ must contain the identity function and all constant functions, and must be closed under composition. Nothing else seems terribly obvious. What is known about this?

Edit: for the purpose of answering the specific questions on MSE, even results limited to connected, bihomogeneous $T_1$ spaces would be very helpful.

An open question on MSE, http://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a topology over that field such that the continuous functions from the field to itself are precisely the polynomials (and whether such a topology exists for $\Bbb Q$, $\Bbb R$, or $\Bbb C$). Another question, linked from there, asks the same about holomorphic functions. The common notion is this:

Given a set $X$ and a set $\mathscr C$ of functions from $X$ to $X$, what are necessary and/or sufficient conditions under which there is a topology $\mathscr T$ on $X$ such that a function $f\colon X\to X$ is $\mathscr T$-$\mathscr T$ continuous iff $f\in \mathscr C$.

It is immediately clear that $\mathscr C$ must contain the identity function and all constant functions, and must be closed under composition. Nothing else seems terribly obvious. What is known about this?

Edit: for the purpose of answering the specific questions on MSE, even results limited to connected, bihomogeneous $T_1$ spaces would be very helpful.

An open question on MSE, http://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a topology over that field such that the continuous functions from the field to itself are precisely the polynomials (and whether such a topology exists for $\Bbb Q$, $\Bbb R$, or $\Bbb C$). Another question, linked from there, asks the same about holomorphic functions. The common notion is this:

Given a set $X$ and a set $\mathscr C$ of functions from $X$ to $X$, what are necessary and/or sufficient conditions under which there is a topology $\mathscr T$ on $X$ such that a function $F\colon X\to X$ is $\mathscr T$-$\mathscr T$ continuous iff $f\in \mathscr C$.

It is immediately clear that $\mathscr C$ must contain the identity function and all constant functions, and must be closed under composition. Nothing else seems terribly obvious. What is known about this?

An open question on MSE, http://math.stackexchange.com/questions/427634/a-topology-such-that-the-continuous-functions-are-exactly-the-polynomials , asks whether there is an infinite field and a topology over that field such that the continuous functions from the field to itself are precisely the polynomials (and whether such a topology exists for $\Bbb Q$, $\Bbb R$, or $\Bbb C$). Another question, linked from there, asks the same about holomorphic functions. The common notion is this:

Given a set $X$ and a set $\mathscr C$ of functions from $X$ to $X$, what are necessary and/or sufficient conditions under which there is a topology $\mathscr T$ on $X$ such that a function $F\colon X\to X$ is $\mathscr T$-$\mathscr T$ continuous iff $f\in \mathscr C$.

It is immediately clear that $\mathscr C$ must contain the identity function and all constant functions, and must be closed under composition. Nothing else seems terribly obvious. What is known about this?

Edit: for the purpose of answering the specific questions on MSE, even results limited to connected, bihomogeneous $T_1$ spaces would be very helpful.