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.

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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 .