Toy Definition of a Topological Space Preamble:
This is a question meant to be fun and amusing -- not part of any class/homework/test/etc.  Also, please forgive my horrific typesetting.  This is my first question on this website.  I'm fluent in Latex, which seems to be close, but not exactly the language structure of this question format.
Motivation:
I was rereading about Point-Set Topology, and was considering to myself the possibility creating an alternative definition to a Topological Space.  To Recall,

Definition: A Topology on a set $X$ is a collection $\tau$ of subsets of $X$ having the following properties:

*

*$\emptyset$ and $X$ are in $\tau$.

*The union of the elements of any subcollection of $\tau$ is in $\tau$.

*The intersection of the elements of any finite subcollection of $\tau$ is in $\tau$.

A set $X$ for which a topology $\tau$ has been specified is called a topological space.    That is, the ordered pair $(\tau,X)$.

-- Topology, Second Edition. James R. Munkres. Page 76.
We see here that $\tau$ is a collection of subsets, as that is what our definition states.  However, suppose instead we imagine replacing $\tau$ with the function $\rho$ which accepts two inputs and provides a boolean output.  To define explicitly:
$\rho: \{\text{ruleset}\} \times \mathscr{P}(X) \rightarrow \{1,0\}$
$\rho(\{\text{ruleset}\},U) = 1$, if $U$ is open or a complement of an open set according to the ruleset.
*notice here that both $X$ and $\emptyset$ would return a value of $1$, as they are both open, and complements of each other.
$\rho(\{\text{ruleset}\},U) = 0$, else.
(We define $\mathscr{P}(X)$ to be the powerset of the set $X$).
Now if the ruleset is something simple such as "If $U$ is in $\mathscr{P}(X)$" then we trivially get back the definition via $\tau$.
If the ruleset includes a proviso for an open subset $A \subseteq X$ so that the intersection of $A$ and $U$ is open provided $U$ is open in $X$, then we get the subspace topology.
Question:  Under this pretense, could we redefine a topological space from $(X,\tau) \rightarrow (X,\rho)$?  I would really would love to see this fail under some special example. >:). Many thanks in advance and happy summer 2021!
 A: Picking on sets that open or the complement of a open set is not going translate nicely into a definition for a topology, as the complement of an open set need not be open (it does however, need to be closed). If you got rid of the "or complement of open set" part, then you've recast the definition of a topology in terms of defining a suitable characteristic function on $\mathcal{P}(X)$.
You may be interested in a somewhat different construct. Topologies could alternatively be defined in terms of what sets are required to be closed instead of which ones are required to be open. I like to call such an object a Cotopology in reference to the natural duality is has with actual Topologies.
Specifically, one can define a Cotopological space $(X,\xi)$ as set $X$ together with a collection of subsets of $X$, $\xi$, such that:

*

*$\emptyset$ and $X$ are in $\xi$.

*The union of any finite collection of sets in $\xi$ is in $\xi$.

*The intersection of any collection of sets in $\xi$ is in $\xi$.

The subsets in the cotopology are exactly those sets that are defined to be closed. Such a space has a natural topology given the collection $\tau =\{ S|S^c \in \xi\}$ where by $S^c$ I mean the complement of $S$ in $X$. Likewise the cotopology can be derived from the topology by the analogous construction (demonstrating the duality between these constructions).
