Consider the letters of the alphabet written as follows:
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
(i) Suppose these letters are written with infinitely thin lines. Classify them according to their hoemomorphism types, i.e., separate them into disjoint sets (equivalence classes), where the members of each set can be continuously deformed into each other. How many equivalence classes do you find?
(ii) Assume that the lines have a certain two-dimensional width and classify the letters again. How many classes can you find now?
(iii) Assume the letters are made from modelling clay so they also have a certain height. How many classes do you find?
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For (i) I've got 9 equivalence classes including {A, R}, {B}, {C, I, J, L, M, N, S, U, V, W, Z}, {D, O}, {E, F, G, T, Y}, {H, K}, {P}, {Q}, {X}. Intuitively, the members in each set can be deformed to each other by stretching, contracting or twisting, imagining that each letter is made by an elastic material. They are also able to be proved homeomorphism by removing key points of each letter.
But for (ii) and (iii), what I can see is just the string in (i) goes to pieces of paper in (ii) and modelling clay in (iii). The change is trivial, since we still deform the letters in the same way (stretching, contracting, twisting)? However if so, this problem would be non-sense, which is not likely to be. Could someone please guide me to the right track and recommend some reading?
P.S. I'm completely a beginner in topology (only got 1 class and there is nothing related in the textbook). So this question might be pretty damn. :) Anyway thanks in advance!!
Let's think about ii).
Consider the letters Q and O. If you draw them as lines then O is a circle and Q is a circle with an extra line segment. As you've correctly identified they're not homeomorphic because different things happen when you remove certain points.
But what if we draw them with a thick pen, so they have a width? Then we can push the nub of the Q inwards and make it disappear. So now Q and O are homeomorphic! Similar things will happen to other letters.