Thanks, Aparna Sathya Murthy , for this useful link.

sorry , its out of my interested area.

See this link

https://math.stackexchange.com/questions/2274743/homeomorphism-between-homeomorphic-spaces

I would not like to add yet another irrelevant comment to your very intriguing question.

Basically homeomorphisms are about continuous deformations preserving "shape" or deep topological properties.

You might look at phase transitions diagrams in thermodynamics, and looking how properties vary when you go a path across elements.

Of course maps and navigation are the historic example of homeomorphisms with angle preservation from a spherical earth to a 2D map. Revisit navigation today, it has lots more to do with topology than just homeomorphisms. In city "canyons", paths are constrained by buildings and Euclidian metrics are irrelevant for determining shortest path. Taxicab (Minkowski) distance with some homeomorphism (LOL, yes!)are much more relevant, to bend shapes from a parallelepidic street model to a real street.

I am sure you can use lots of topology there. By the way pedestrian navigation is still very poor, probably due to a macrofactor that the pedestrian is not welcome in the USA, where people have four wheels more often than two legs (lack of homeomorphism between two legs and four wheels!). Therefore dominant navigation platforms made in USA considered pedestrians as a second thought, bad bad bad!

Maybe this is too superficial a consideration, but it has some truth.

Pedestrian navigation should start from walking: two legs, one brain, one body to carry. Homeomorphic streets which I call city canyons, and buildings. Graphs after thinning the network of interconnected streets (Homeomorphic modeling).

There is an interesting homomorphism in physics: from phase to wave. It's a group homeomorphism rather than a topological one.

Say Phase = omega.t - k. x - phi

Then wave = f(Phase) with f comprising an exponential exp(i.Phase) to describe an oscillator for instance, or a non-imaginary part for an amortissement tending to zero at infinity.

Of course exp(a+b) =exp(a). exp(b) hence an additive group maps into a multiplicative group.

I wonder if it's not more powerful to look for invariants or conservation equations. Homeomorphisms are "shape essence" preserving, this is a kind of invariance, but it's very restrictive.

It would be interesting to look at non-homeomorphic systems coming from the physical world. Imagine a Gram-Schmidt matrix (which can be seen as an autocorrelation), which diagonalises, and maybe as a control parameter lambda evolves, some eigenvalues which could have been identical initially, split, and you get different eigenvalues, thus physically different resonance modes. It's like a guitar with N times the same chord, evolving into N chords guitar (non Homeomorphic....)

You may look again at Homeomorphic systems evolving away from homeomorphy, as the fixed points do in the Feigenbaum cascades.

I like disruptions more than invariance...

Article Journal of Mathematical Acumen and Research 2(2): 1-4, 2017 ...

Thanks, Aparna Sathya Murthy , for this useful link.

Thanks for the stimulating question!

Thanks Dr. Renaud Di Francesco , I appreciate the interesting and valuable explanations you have presented.