I have found this exercise question in the book "Planning algorithms - Steven M. LaValle". I did not get how we categorize some pairs as homeomorphic?
For the letters of the Russian alphabet, А, Б, В, Г, Д, Е, Ё, Ж, З, И, Й, К, Л, М, Н, О, П, Р, С, Т, У, Ф, Х, Ц, Ч, Ш, Щ, Ъ, Ы, Ь, Э, Ю, Я , determine which pairs are homeomorphic. Imagine each as a 1D subset of R2 and draw them accordingly before solving the problem.
It's a little misleading to talk about pairs of homeomorphic letters when the more natural concept would be to put them in equivalence classes.
Intuitively, if these letters are made of rubbery strings, and we can transform them by stretching but not cutting or gluing, then which ones are related by these transformations?
For example, one of the equivalence classes is $\large \{\textsf {Б, Р, Ъ, Ь}\}$; this is the class of letters having a loop with a line sticking out.
Another class is $\large \{\textsf {Г, З, И, Л, М, П, С}\}$, containing all the letters which are homeomorphic to a straight line. (Note that $\large \textsf T$ and $\large \textsf Ш$, for example, do not belong to the class because they have a junction.)
Another class is $\large \{\textsf Ё\}$, which contains all the letters consisting of three disjoint pieces.
I'll let you do the rest. As a final comment, observe that I've used a sans-serif font for the letters. This makes a difference, because $Б$ and Б are not homeomorphic!
Here is the sans-serif Cyrillic alphabet as a picture, in case the text above doesn't display properly:
