By Stephen Huggett

This is a booklet of uncomplicated geometric topology, during which geometry, usually illustrated, publications calculation. The booklet starts off with a wealth of examples, frequently sophisticated, of the way to be mathematically yes no matter if items are an identical from the viewpoint of topology.

After introducing surfaces, similar to the Klein bottle, the e-book explores the houses of polyhedra drawn on those surfaces. extra sophisticated instruments are constructed in a bankruptcy on winding quantity, and an appendix offers a glimpse of knot concept. additionally, during this revised version, a brand new part provides a geometric description of a part of the class Theorem for surfaces. numerous notable new photographs exhibit how given a sphere with any variety of traditional handles and no less than one Klein deal with, all of the traditional handles could be switched over into Klein handles.

Numerous examples and workouts make this an invaluable textbook for a primary undergraduate direction in topology, offering a company geometrical starting place for additional learn. for far of the ebook the must haves are mild, even though, so someone with interest and tenacity may be in a position to benefit from the *Aperitif*.

"…distinguished by means of transparent and beautiful exposition and encumbered with casual motivation, visible aids, cool (and fantastically rendered) pictures…This is a very good e-book and that i suggest it very highly."

MAA Online

"*Aperitif* inspires precisely the correct influence of this publication. The excessive ratio of illustrations to textual content makes it a brief learn and its enticing variety and subject material whet the tastebuds for more than a few attainable major courses."

Mathematical Gazette

"*A Topological Aperitif* presents a marvellous advent to the topic, with many alternative tastes of ideas."

Professor Sir Roger Penrose OM FRS, Mathematical Institute, Oxford, united kingdom

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**Extra info for A topological aperitif**

**Example text**

Perhaps the reader feels that the Klein bottle has been treated unsatisfactorily vaguely in comparison with the sphere and the torus. We are about to look at the Klein bottle in other ways, both simple and precise, but not without the penalty of moving away from our hitherto down to earth approach in which every set is Euclidean. However, if the reader prefers to stay in the safe universe of these Euclidean sets, it would be very elementary, although somewhat tedious, to give a version of our construction using a tube of square cross section, so that the Klein bottle was made of ﬂat pieces each given by a straightforward formula.

Whatever a topological space is, we would still like it to make sense to ask whether two topological spaces are homeomorphic or not. But being homeomorphic depends on continuity, which in turn depends on the idea of neighbourhoods: a mapping f from a space X to a space Y is continuous at the point x ∈ X if the pre-image of N is a neighbourhood of x whenever N is a neighbourhood of f (x). Consequently we deﬁne a topological space to be a set X with a speciﬁcation of the neighbourhoods—subsets of X containing x—of each point x of X.

5 it follows that f sends close components to close components and f −1 sends close components to close components, so that the closeness relations of X and Y are isomorphic. This completes the proof. 29, if these were equivalent, their complements would be equivalent and would have isomorphic closeness graphs. But the tree for X has a vertex related to three others, whereas the tree for Y does not, so the two trees are not isomorphic. Take, for example, the case of putting ﬁve circles in the sphere.