Even 'topology' is a little confusing to those more familiar with the term from maths.
For the 'CATH' hierarchical classification, the 'Topology' level is something like the organization of secondary structure in an 'Architecture'. This has some relationship to topology in the general sense, but is a narrower definition.
For me, the 'fold' is what happens after 'folding' occurs, but I take the point that it is confusing.
Topology actually makes some sense here in that a very small number of proteins do fold into knots! This was a huge surprise and completely contradicted most predictions. https://en.wikipedia.org/wiki/Knotted_protein
knots in the real world (actual, real knots) don't have tied-together ends either. That's not what a knot is. The study of topology in math works on abstract math-knots.
Generally, any non-covalent bonds aren't considered to be topological connections in proteins, although salt bridges can definitely ahve bond energies not far from covalent bonds.
For the 'CATH' hierarchical classification, the 'Topology' level is something like the organization of secondary structure in an 'Architecture'. This has some relationship to topology in the general sense, but is a narrower definition.
For me, the 'fold' is what happens after 'folding' occurs, but I take the point that it is confusing.