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**Example text**

Consequently S \ X and S \ Y are equivalent in S. This completes the proof. 2 gives the following result. 3 Equivalent subsets have homeomorphic complements. 7 34 A Topological Aperitif We now give three plane sets, each homeomorphic to the open interval ]0, 1[, that are non-equivalent subsets in the plane. Let X be ]0, 1[, let Y be the unit circle with its north pole removed, and let Z be the whole real line. The complement of Z is not path-connected, whereas the complements of X and Y are. Also the complement of Y has a cut-point, namely the north pole, whereas the complement of X does not.

To prove the general result that any two (m, n)-circles are equivalent in T we show that the (m, n)-circle is equivalent to the (0, 1)circle. Deﬁne the mapping g from the plane to itself by (u, v) → (tu + mv, −su + nv), where s, t satisfy ms + nt = 1: we are assured 3. Equivalent Subsets 33 that such s, t exist as m, n are relatively prime. As all four coeﬃcients of u, v in the formula for g are integers, g does give a mapping T → T, which is continuous and indeed a homeomorphism as the inverse of g is (u, v) → (nu − mv, su + tv).

Find eleven examples of such a set S, no two being homeomorphic. Show that no two of your sets are homeomorphic. 5. Let L1 be the set {(x, 0) : 0 ≤ x < 1} ∪ {(0, y) : 0 ≤ y < 1} and let L2 be congruent to L1 . The plane set T is L1 ∪ L2 . Sketch eight examples of such a set T, no two being homeomorphic. Show that no two of your examples are homeomorphic. 6. 5 have points where three lines emanate: in fact S has one whereas T has two. Give a precise deﬁnition of an n-node, a point where n lines emanate, and show that a homeomorphism sends an n-node to an n-node.