I noticed that there are two different things in mathematics both referred as “generalization”.

The first is like replacing real numbers with complex numbers, that is replacing a set in consideration with its superset.

The second is like replacing a metric space with its topology, that is abstracting away some properties.

Why are both called with the same word “generalization”? What is common in these two? Please comment.

One path to generalization passes from a smaller space to a larger space.

Another path to generalization passes from more axioms to fewer axioms.

They sometimes converge, sometimes not.

I tried making this point on MathOverFlow once but it was not well received.

On a related note, C.S. Peirce recognized two types of abstraction —

• Prescisive Abstraction

• Hypostatic Abstraction

Isn’t the “set” of topological spaces a “superset” of the “set” of metric spaces?

It isn’t. Multiple metric spaces correspond to a single topology not vice versa.

I think mathematicians outside category theory often treat structures as if they were properties. For example, they treat metric spaces as just “metrizable topological spaces”. In that regard, then forgetting a structure is kind of like generalizing to the superset which may not have that selected property.

It seems to me that category theorists have made this quite precise here: https://ncatlab.org/nlab/show/stuff%2C+structure%2C+property .