You might think that the title is an overstatement of a well-known fact but it is the best title I can come up with for the wonders the intersection operator does in some fields of math.


Recently,(on summer vacation) I was studying one subject after another and after changing about three subjects, I began to notice that in all these the set-theoretic intersection operator always carried over some property of the parent sets to the one obtained after the intersection.


To summarize briefly:



> 
> Let $A$ and $B$ be two sets, say with property P. Then, $A \cap B$ has property P.
> 
> 
> 


Evidences, some trivial:


Topology
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* The intersection of two open sets is open.
* The intersection of two closed sets is closed.
* The nonempty intersection of two subspaces of a metric space is a metric space.


......and so forth.


Algebra
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* The intersection of two subspaces of a vector space is a vector.
* The intersection of two subgroups of a group is a group(w.r.t the same binary operation and clearly the intersection is between the underlying sets).
* The intersection of two sub-fields of a field is a field.


......the list continues.


The third subject was Graph Theory, but I haven't yet come across the notion of intersection.


Now I would like to ask ~~whether this trend always holds or~~ whether there is some underlying principle each discipline abides by when using the notion of intersection. ~~Is there any property deviating from this trend?~~ What are the reasons for the ubiquity of the quoted property?



The unprecedented success of the “intersection” operator
When property P is universal ($\forall ...$) it is likely to correspond to closed sets, and thus be preserved under intersection. Examples: axioms of a group, ring, field, directed graph; having symmetry under a given group. 


However, if P is existential ($\exists ...$) it corresponds to open sets and is more likely to be preserved by unions (or products), not intersections. Examples: being algebraically closed, having at least 53 elements. (Well, algebraic closure is $\forall \exists$ so of course it is even more complicated. But falling out of the pure $\forall$ class it fails the intersection property.)


The first situation is possibly more common because we want structures to satisfy some, well, structural properties. Properties expressed by equations usually correspond to closed sets.


To some extent this is formalized in Birkhoff's theorem On equational presentations. Any book on Universal Algebra will discuss it.


Also, the sample of concepts is biased, because definitions that become standard are often selected for their useful formal properties. Concepts not having stability under intersection (or union, or inheritance by sub- or super-structures) are less likely to be used.


