Is there any study of putting a topology on a topology. In other words..
In topology, we start with a set $X$ and define a topology $\tau\subseteq 2^X$ satisfying certain axioms (where 2^X denotes the power set of $X$). But the set $X$ was arbitrary, so it is possible that $X$ could happen to be a topology on some other set (though, not on itself).
Conversely, say we have a set $X$ and a topology $\tau$ on $X$. Can we define a topology $\tau'$ on $\tau$? .... well, yes! Of course we could always pick the discrete or indiscrete topology... but that doesn't seem very interesting. However there are other potential topologies using the fact that $(\tau,\subseteq )$ is a partial order. For example, for any $U\in \tau$ define $U\downarrow\in 2^{\hspace{.06cm}\tau}$ by $$ U\downarrow=2^U\cap \tau=\{V\in \tau\mid V\subseteq U\}$$ Using the fact that $(U\cap V)\downarrow=(U\downarrow)\cap(V\downarrow)$, one easily checks that the set $$\mathcal{B}(\tau)=\{\hspace{.3cm}U\downarrow\hspace{.2cm}|\hspace{.3cm} U\in \tau\hspace{.3cm}\}$$ is a basis for a topology on $\tau$. Let's call that the downward topology on $\tau$, and denote it $D(\tau)$.
Now suppose we do this process iteratively. That is, we are given some topological space $(X,\tau_1)$, and for all $n\geq 1$ define $\tau_{n+1}=D(\tau_n)$. Then we have a sequence of sets $(X=\tau_0,\tau_1,\tau_2,\tau_3,\ldots)$ such that $\tau_{n+1}$ is a topology on $\tau_n$ for all $n\geq 0$. Let's call such a sequence of sets an $\infty-$topological space, or just an $\infty$-topology.
Ok, so what's the point of this? Let's try an example. If we start with $\tau_0=\varnothing$ and define $\tau_{n+1}=D(\tau_n)$, then we get \begin{align*}\tau_0&=\varnothing\\\tau_1&=\{\varnothing\}=\{\tau_0\}\\\tau_2&=\{\varnothing,\tau_1\}=\{\tau_0,\tau_1\}\\\tau_3&=\{\varnothing,\{\tau_0\},\{\varnothing, \tau_1\}\}=\{\tau_0,\tau_1,\tau_2\}\\&\vdots\\\tau_{n+1}&=\{\tau_0,\tau_1,\cdots,\tau_n\}\end{align*}
So if we use the von Neumman definition of the natural numbers (i.e. axiom of infinity of set theory), then we see that $\tau_n=n=\{0,1,2,\ldots,n-1\}$. That's at least a little interesting! The regular old natural numbers $(0,1,2,3,4,\cdots)$ are a non-trivial example of an $\infty$-topological space! (technically, the identity function $\mathbb{N}\to\mathbb{N}$)
There's also the trivial examples where we define $\tau_{n+1}=2^{\tau_n}$ or $\tau_{n+1}=\{\varnothing,\tau_n\}$ to be discrete or indiscrete, respectively. But notice that there are $2^{\aleph_0}$ ways to pick an $\infty$-topological space where at each step we pick $\tau_{n+1}$ to be either discrete or indiscrete. Suffice it to say, there are lots of examples...
Ok, now that we know there exist some examples, what do we do with them? Well, let's follow the pattern of regular topology. One of the basic definitions is what is means for a function $f:(X,\tau_X)\to (Y,\tau_Y)$ to be continuous. We say $f$ is continuous if $U\in \tau_Y\Longrightarrow f^{-1}(U)\in \tau_X$. In other words, the induced function $f^{-1}:2^Y\to 2^X$ restricts to a well-defined function $f^{-1}:\tau_Y\to \tau_X$. Now, the question is: is that function itself continuous? For that question to make sense we need to have topologies on $\tau_X$ and $\tau_Y$....
Suppose $(X_0,X_1,X_2,\ldots)$ and $(Y_0,Y_1,Y_2,\ldots)$ are $\infty$-topologies, and let $f:X_0\to Y_0$ be a function. Then it seems reasonable to define $f$ to be $\infty$-continuous if \begin{align*}f:X_0\to Y_0\\f^{-1}:Y_1\to X_1\\f^{-2}:X_2\to Y_2\\f^{-3}:Y_3\to X_3\\f^{-4}:X_4\to Y_4\\\vdots\end{align*} are all well-defined functions (Here the notation $f^{-(n+1)}$ just means the pre-image under the function $f^{-n})$.
One could also talk about induced topologies. Suppose we have some given $\infty$-topology $X=(X_0,X_1,X_2,\ldots)$ and a set $Y_0$ which we want to make into an $\infty$-topology. If we are given a function $f:X_0\to Y_0$, then we can defined $Y_1$ to be the topology co-induced by $f$, $$Y_1=\{U\subseteq Y_0\mid f^{-1}(U)\in X_1\}$$
Now we have a function $f^{-1}:Y_1\to X_1$, where $X_1$ already has the topology $X_2$. So we can define $Y_2$ to be the topology induced by $f^{-1}$: $$Y_2=\{f^{-2}(U)\mid U\in X_2\}$$
Continuing in this fashion gives an $\infty$-topology $Y=(Y_0,Y_1,Y_2,\ldots)$ such that $f:X\to Y$ is $\infty$-continuous. Likewise, we could extend many of the usual constructions/definitions from topology to $\infty$-topologies.
I guess my overall question is: Is this at all studied? I have tried to find references about it and can't find anything. Would this generalization be useful? Or would the same information be representable in terms of regular topologies somehow?
