I am a bit confused with topology and am reading the next section after configuration spaces. I bumped into the question and got really confused.
just google "topology equivalence" https://www.tandfonline.com/doi/abs/10.1080/00029890.1960.11990308?journalCode=uamm20
Basically topology equivalence means, that there is some 1-1 correspondence continuous in both ways.
That mean, that exist some correspondence, here for point "near" point A on first space are corresponding points on second place "somehow near" point corresponding A.
Basically it means, that all object on one site have corresponding objects on other site and number of "holes" and "islands" in such holes and holes in that islands and ... are the same in corresponding objects. Nothing more.
Basically triangel, circle, square ... of any sizes, are topologically equivalent, as they are "single objects without holes"
The same goes for any cubes, balls, dodecaeders ... regardless size.
If you wil make the object from something, witch could be infinitelly strenchable and can it somehow streng and bend to the other shape, but without connect it or tear it anywhere, then the original and results are topologically equivalent