By Andrew J. Casson

This ebook, which grew out of Steven Bleiler's lecture notes from a path given by way of Andrew Casson on the college of Texas, is designed to function an creation to the purposes of hyperbolic geometry to low dimensional topology. particularly it offers a concise exposition of the paintings of Neilsen and Thurston at the automorphisms of surfaces. The reader calls for purely an knowing of simple topology and linear algebra, whereas the early chapters on hyperbolic geometry and geometric constructions on surfaces can profitably be learn via somebody with an information of ordinary Euclidean geometry meaning to research extra abour different 'geometric structures'.

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Proof: 91 = {M t } is a basis of M\. 6, {U M} is therefore a basis of V M~?. Hence U M, = V M^and V ^ i s a principal filter. 4 Π M t < M i 0 for all indices i0; hence Π Mt ζζ. 6 {Μλ n ... n M„} is a basis of Mx A ... Λ Μ η , whence M1 n ... n M ^ = A/j" Λ ... Λ Μ^. I The following example shows that the inf of an infinity of principal filters need not be a principal filter: Let An be the set of natural numbers ^ n. Then H {^4W : ne N} = 0, hence Π ^4n = 0 = 0. However, the filter /\ An is the Fréchet-filter associated with the natural numbers (cf.

From S eUv and q e 6 , it follows that S eUQ and hence (U 3 ) is verified with U = V = S. 6 we see that rp = Uv defines a topology τ on R. From S e S and p e S we have S eUP , hence S G rp. T h u s S contains only r-open sets. On the other hand, let M be an arbitrary r-open set. 1, an Sv e S such thâtp G Sv and Sv c M. By (c) we likewise have M = U {Sv : p G M} e S This proves that 6 contains all possible τ-open sets. To prove the converse, let r be a topology on R. 3 the system 6 of all r-open sets possesses properties (a)-(c) and thus defines, in the manner pre viously shown, a topology τ' on R.

Since R contains all points, it also is closed. But Co = R and CR = & and both these sets must therefore also be open. , Mn be open sets and pe Μλη ... n Mn . Then pe Mv and (Definition 3a, FJ. , n. Therefore Mxn ... n Mnerp 2 A point p such that M e rp is also called inner point of M. A set is open if and only if all its points are inner points. 3 When we wish to emphasize the underlying topology r without which, of course, we could not define our new concepts, we speak of "τ-open" and "τ-closed" sets, etc.