Why are the Kuratowski closure axioms so interesting? I never really understood the relevance of the Kuratowski closure axioms. My problem is this:

A user2520938 closure operator is an assignment $cl:\mathcal{P}(X)\to\mathcal{P}(X)$ s.t.
  
  
*
  
*$cl(X)=X$ and $cl(\emptyset)=\emptyset$
  
*For any collection of sets $U_\alpha\in \mathcal{P}(X)$ s.t. $cl(U_\alpha)=U_\alpha$ for all $\alpha$ we have $cl(\bigcap U_\alpha)=\bigcap U_\alpha$
  
*For any finite collection of sets $U_i\in \mathcal{P}(X)$ s.t. $cl(U_i)=U_i$ for all $i$ we have $cl(\bigcup U_i)=\bigcup U_i$
  

Then this will clearly also induce a topology by saying $U$ is closed iff $cl(U)=U$.
Now of course it's a bit silly to specify the operator in this fashion. However, this does make me wonder what's so special about the formulation of the Kuratowski closure axioms that they deserve a name and fame?
I suspect maybe the reason is that the Kuratowski formulation of the axioms makes it easy to verify that a given operator is a closure operator, since every axiom involves only $1$ or $2$ sets, while 'my' formulation here requires a lot more work?

A Kuratowski closure operator $f:\mathcal P(X)\to\mathcal P(X)$ has the property that there is a (unique) topology $\tau$ on $X$ such that $f$ is the closure operator for that topology, i.e., for every subset $U$ of $X$, $f(U)$ is the $\tau$-closure of $U$.
A "user2520938 closure operator" $f:\mathcal P(X)\to\mathcal P(X)$ in general will not have that nice property. For example, if $(X,\tau)$ is a topological space, and if I define  $f:\mathcal P(X)\to\mathcal P(X)$ by setting $f(U)=U$ if $U$ is $\tau$-closed and $f(U)=\emptyset$ otherwise, then $f$ is a "user2520938 closure operator".
In other words, your conditions, besides being more complicated, are insufficient to characterize topological closure operators.
