Tweaking the axioms of a Topological Space, what are the consequences? A topological space is a set $X$ together with a topology $\tau$ (a collection of open subsets) such that. 


*

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

*The intersection of a finite number of sets in $\tau$ is also in $\tau$. 

*The union of an arbitrary number of sets in $\tau$ is also in $\tau$. (it could be the union of infinitely many sets)


what if we tweak this definition so that it becomes:

A topological* space is a set  a set $X$ together with a topology* $\tau$ (a collection of open subsets) such that. 
  
  
*
  
*$\emptyset\in \tau$ and $X\in \tau$. 
  
*The intersection of an arbitrary number  of sets in $\tau$ is also in $\tau$. 
  
*The union of a finite number of sets in $\tau$ is also in $\tau$.
  

how will that change how the concept of a topology captures that of a neighborhood intuitively speaking? (this is the most important to me) how are these topological* spaces going to differ from topological spaces? was this particular definition ever historically coined down?
thx in advance for any kind of help
 A: If $\{C\}$ is a collection of sets satisfying the requirements of your tweaked definition, then $\{X-C\}$ is an ordinary topology in the original sense. In other words, your "tweaked" definition produces the closed sets of some topology consisting of open sets in the ordinary sense.
A: It is clear that items 1 and 3 in your enumeration are included in (implied by)  the original definition of topological   space; hence these are natural properties that a “neighborhood” should have. So the only point to discuss is whether lifting the restriction of finite intersections takes somewhere counterintuitive.

Remark. Neighborhoods of a point $x$ are usually sets $V$ such that there exists an open $U$ with $x\in U \subseteq V$, so one should take into account that the intuition is about subsets that are neighborhoods of all of their points.

A first example: The discrete topology $\mathcal{P}(X)$ on any $X$ satisfies both the axioms for a topology and yours. This is a first hint that anything one can say should be based in some strong picture of neighborhood, like that of a (locally) connected space as $\mathbb{R}^n$. Even “metric space” is not enough, since the discrete topology is metrizable.
Take, for instance, the concept of a limit of a sequence. We would like to say that $x_n\to x$ as $n\to\infty$ whenever the tail of the sequence $\{x_n\}_{n>N}$ is “closer” to $x$ as $N$ grows. That would mean, taking  neighborhoods $U_1\supset U_2\supset U_3\supset \dots \supset U_m\supset \dots$ of $x$, all but finitely many terms of the sequence lie in $U_1$, and the same is true for the other $U_m$. If you allow arbitrary intersections,  there must be a smallest neighborhood of $x$, and hence in a topological* space the sequence gets infinitely close to $x$ in a finite time. That is, there is an $M$ such that for all neighborhoods $U$ of $x$ and all $n>M$, $x_n\in U$.
If in your intuition of space, you may approach a point without never being “infinitely close” but getting every time closer, you can't have arbitrary intersections of neigborhoods.
EDIT: As Paul Sinclair observed in the comments, allowing topologies* that are not closed under arbitrary union also goes against some intuitions. This follows easily from the highlighted remark above, plus the observation that (using that terminology) if $V$ is a neighborhood of $x$ and $W\supseteq V$, then $W$ also is. This automatically implies closure under arbitrary unions.
