[Math] Meaning of “a mapping preserves structures/properties”

Question

1. 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?

2. I also see that "a mapping preserves some property of subsets,
   points or whatever". Such as

   Continuous linear mappings between topological vector spaces preserve
   boundedness.

   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!

Answer 1

Top is the category of topological spaces and continuous maps simply by definition; topology typically deals with continuous maps, making this category the most important one, and thus by convention it's the one meant when saying "the category of topological spaces".

(aside: other conventions on what Top or "the category of topological spaces" stands for are far more likely to disagree on what the objects are, rather than the morphisms. e.g. to make the objects be merely the compactly generated Hausdorff spaces)

You can, of course, make all sorts of other categories. The category of topological spaces and open maps is a perfectly reasonable category to make; it's just less useful.

It takes a bit to get used to, but category theory rejects the mindset that mathematics is about objects, with the mappings between objects being a derived notion. Instead, you need to consider objects and mappings as equals -- or even to consider the objects superfluous.

On that last point, my favorite example of a category whose emphasis is on the morphisms is matrix algebra. The set of all matrices, with composition defined by multiplication, form a category. (with addition, you get an Abelian category) The objects of this category really play no role beyond bookkeeping to say which matrix products are defined.

(This category is, of course, equivalent to the category of fintie-dimensional vector spaces and linear maps)