Download Topology: A Geometric Approach by Terry Lawson PDF

By Terry Lawson

This new-in-paperback creation to topology emphasizes a geometrical strategy with a spotlight on surfaces. a first-rate function is a big choice of workouts and initiatives, which fosters a educating kind that encourages the coed to be an energetic type player. a variety of fabric at assorted degrees helps versatile use of the e-book for various scholars. half I is acceptable for a one-semester or two-quarter path, and half II (which is challenge dependent) permits the publication for use for a year-long direction which helps numerous syllabuses.

The over 750 workouts diversity from basic exams of passed over information in arguments, to enhance the fabric and elevate scholar involvement, to the improvement of considerable theorems which have been damaged into many steps. the fashion encourages an energetic scholar function. recommendations to chose routines are integrated as an appendix, with options to all routines to be had to the teacher on a better half web site.

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Extra resources for Topology: A Geometric Approach

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The only missing ingredient from our proof above is knowing that a cube [a1 , b1 ] × [a2 , b2 ] × · · · × [an , bn ] is compact. This can be proved inductively if we can show that the product of compact sets in a product of Euclidean spaces is compact. To do this most efficiently, we need to discuss the notion of a product topology on the product X × Y of two topological spaces. Suppose that X and Y are topological spaces and consider their product X × Y = {(x, y): x ∈ X, y ∈ Y }. We will define a topology on X × Y by saying that a set W ⊂ X × Y is open if given any (x, y) ∈ W , then there are open sets U in X and V in Y so that (x, y) ∈ U × V ⊂ W .

Show that two disjoint concentric circles in the plane are not homeomorphic to one circle. 7. 7 37 Quotient spaces Quotient spaces We discuss the notion of a quotient space, which is also called an identification space. We will be using quotient spaces extensively in Chapter 2 when we study surfaces. 1. Suppose X, Y are topological spaces, and we have a surjective map q : X → Y . Then we say Y has the quotient topology with respect to (X, q) if U ⊂ Y is open iff q −1 (U ) ⊂ X is open. Y is then called a quotient space of X and q is called a quotient map.

Thus any two triangles in the 18 1. Basic point set topology plane are homeomorphic via a canonical affine linear map, and the image of a triangle under an affine linear map will be another triangle. In particular, there is no affine linear map sending a triangle to a rectangle. Affine linear maps from one triangle ∆(a0 , a1 , a2 ) to another triangle ∆(b0 , b1 , b2 ) are determined completely by the map on the vertices ai → bi and the affine linearity condition λi ai → λi ai . 3. (a) Show that a1 − a0 , a2 − a0 are linearly independent iff λ0 a0 + λ1 a1 + λ2 a2 = 0, λ0 + λ1 + λ2 = 0 implies λ0 = λ1 = λ2 = 0.

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