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:
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.
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?