2026-09-04

Hierarchy of Mathematical Spaces

As you can see in 20260904T164816-connection_bw_topology_and_abstract_algebra and 20260904T134153-topological_scaffolding there is a natural(? We created it but things neatly map so it is sort of natural) hierarchy amongst mathematical objects.

Qef, Public domain, via Wikimedia Commons

These are not inherited. It is actually the other way around.

All Inner products can induce a valid norm All norms can induce a valid metric All metrics can induce a valid topology

It grows from the inside. Though it was introduced as adding length to X gives you Y, I think it is actually Y having length also makes it a X.