BEGINNING TOPOLOGY
NOTES and ERRATA (updated 10/22/07)
Page 7 Exercise 1.1.20:
In ternary, 0.1 =
0.0222. . . (not 0.222. . . )
Page 9 Example 1.1.27: ~ should be defined as
(0, y) ~ (1, y) for all y in [0, 1],
and for any point (x, y) in X, (x, y) ~ (x, y) (hence reflexive)
and if (x1, y1)
~(x2, y2) then (x2, y2) ~ (x1,
y1) (hence symmetric—this is the correction)
Page 28 The open cover definition of compactness should be expanded to subspaces. Add: “A subspace Y of X is compact if every family of open subsets of X whose union contains Y has a finite subfamily whose union also contains Y.”
Page 62 Step 5:
In Cut 2: note that the “beginning” of x-1 is the “end” of x.
In Cut 3: This would be clearer as “Cut from the end of y to the end of x (call this cut z). . .”
Page 63 Figure 2.35(b): The dotted curve labeled y should end at the vertex (not along the side labeled x).
Figure 2.35 (c): Similarly the dotted curve labeled z should begin and end at vertices.
Page 79 Since a k-connected sum does not give a unique surface (the orientation depends upon how the disk boundaries are glued), change exercise 3.3.6 change to read : . . . define a “k-connected sum”. . .
Page 93 last line, add
period at end.
Page 98 Figure 4.7: The figure is slightly off. The face A should meet the top and bottom of the square in corresponding intervals. Same for faces B and C.
Page 102 line 7: Omit “We in fact disallow them in this section.”
Page 102 lines 8-9, Omit “Assumption.” since parallel edges are allowed, for example, in Euler’s theorem.
Page 107 Project 4.3.11: The goal is to arrange the cubes in a column so that each side of the column shows all four colors (that is, no two faces are the same color).
Page 110 Note that
although Kuratowski’s theorem is correctly
stated, graph homeomorphism has not been defined, so a better statement is “A graph is planar if and only if it
cannot be transformed into K5 or K3,3 by a series of
edge contractions and deletions, and vertex deletions.”
Page 110 last paragraph, correct to Erdős
Page 124 Figure 5.9 right side: the arrows are
incorrectly drawn near the critical point. They should keep circulating around
the critical point.
Page 132-133, proofs of theorems 5.3.5 and the Poincare-Bendixson theorem. An additional sentence might be
helpful to explain why an orbit cannot turn around to meet S again in the opposite direction: “Note that all orbits must meet S
with the same orientation since S is never tangent
to an orbit.”
Page 136 last paragraph and page 137 first paragraph should read:
“Draw a curve C, made up of pieces of
orbits and pieces orthogonal to orbits, around a nonrotational
critical point p so that C encloses no other critical points. To simplify
counting the angle variation, we will assume that the separatrices
bounding the various sectors are straight line segments.
Suppose we have E elliptic sectors, H
hyperbolic, leaving P parabolic sectors as the remainder. Let a1, . . ., ae be the angles between the separatrices
bounding elliptic sectors. Similarly let g1, ...,gh be the angles of the
hyperbolic sectors and b1, ..., bp the remaining parabolic. Figure 5.25
illustrates the subdivision into sectors and the associated angles for a
critical point with sectors of each type.”
Page 137 Figure 5.25 should then be as follows:
Page 138 line 4: (g1 – p) should read (g1 - p)
Page 147 last 2 lines—Note that one can
define the index of a critical point on a nonorientable
surface by the formula in theorem 5.4.3 and use an oriented double cover of the
surface.
Page 199 and page 232 M. Thistlethwaite’s name is misspelled. My apologies.
Page 224 The Kauffman polynomial as defined here is more properly referred to as the Kauffman’s “X-polynomial” or the “normalized bracket polynomial” to distinguish it from the more powerful 2-variable invariant polynomial F_{K} as defined in Kauffman’s TAMS paper "An Invariant of Regular Isotopy"