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

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

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?

notsubgroups does not necessarily have that same property... – Mariano Suárez-Alvarez♦ Jul 26 '10 at 7:59