Download A Survey of Geometry (Revised Edition) by Howard Eves PDF

By Howard Eves

From the book's preface:
Since writing the preface of the 1st variation of this paintings, the gloomy plight there defined of starting collegiate geometry has brightened significantly. The pendulum turns out certainly to be swinging again and a goodly volume of good textual fabric is showing.

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Extra info for A Survey of Geometry (Revised Edition)

Example text

On the other hand, if p: M •* B is merely a continuous surjection, one can presumably adapt the modified compact-open topology of Booth and Brown (1978) to make inversion II(M) + II(M) , ? K £; continuous and thus, under some suitable local compactness condition on M, make II(M) into a topological groupoid, even when M ->• B has no local triviality properties. This is the more interesting of the two generalizations, but we have no specific need for it. See also Seda (1980, §4). 14. // The following example is from Brown and Danesh-Naruie (1975).

This is the more interesting of the two generalizations, but we have no specific need for it. See also Seda (1980, §4). 14. // The following example is from Brown and Danesh-Naruie (1975). 25 Let B be a path-connected, locally path-connected and semi-locally simply connected space. The first condition ensures that the fundamental groupoid /[(B) is transitive; sets U the last two that the topology of B has a basis of open, path-connected such that the inclusion U £ to the trivial subgroup of TT (B,x).

Then ft' and <(> are (isomorphic to) the produced groupoid and produced morphism of ft along , . D Proof. 21. 21). 10 that if ft is locally trivial and :ft+ ft1 is a base-preserving morphism then ft1 is locally trivial and, what is more, any open cover {U } which is the domain of a cocycle for ft will also be the domain of a cocycle for ft*. Loosely speaking, ft* can be no more twisted than ft is (and this is so even though need not be onto; indeed need not even be a morphism - one needs only that afo

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