Filters and Ideals are dual.

There is a duality between bornology and uniform structures (in Weil's sense). If E is a bornological space and F a uniform space, the space F^E of mappings from E to F carries a natural uniform structure (uniform convergence on bounded sets) ; if F and G are uniform spaces, the space C(F,G) of uniformly continuous mappings from F to G carries a natural bornology (the set of equicontinuous sets of mappings). The relation becomes more significant for topological and bornological vector spaces

Thanks profesor Houzel.

Bornology is an ideal of sets satisfying another condition: this ideal covers the whole space.

Bornology is a kind of dual to  topology.

Dear prof arthur can u explain in details plzz

Yes a bornology is an ideal right and ideal and filter are dual structures.

One way to see a duality  is probably to notice that a morphism in Top is defined by: the preimage of the target topology is in the source topology. In Bor a morphism is defined by: the image of the bornology in the source is in the bornology in the target.

Bornology is a structure built with a use of a covariant functor.

Topology is constructed by a use of a contravariant functor.

i understand very well in categorical language bornology is covariant onthe other hand topology is contravariant thanks..

they are dual each other