Sometimes I see something like "a mapping preserves the structures of its domain and of its codomain". From Wiki about morphisms in category theory:
a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.
I was wondering why the structure-preserving mappings between two topological/measurable spaces are defined by the "inverse" of the mapping, while the structure-preserving mappings between two groups/vector spaces are not?
Why are the structure-preserving mappings between two topological spaces chosen to be continuous mappings instead of open mappings?
I also see that "a mapping preserves some property of subsets, points or whatever". Such as
According to Brian's reply to my earlier question, this quote should be understood as "under a continuous linear mapping, the image of any bounded domain subset is also a bounded codomain subset", not as "under a continuous linear mapping, the inverse image of any bounded codomain subset is also a bounded domain subset".
I wonder why? It seems at first to me like how continuous mappings preserve topologies, but it is actually in the same way as how group homomorphisms preserve group structures.
Thanks and regards!