John Stillwell - Naive Lie Theory.pdf - Index of

9.5 Deforming a path in a sequence of small steps 195
of Section 8.5. In other words, for each ε > 0 there is a δ > 0 such that
|p(Q)− p(R)| < ε for any points Q,R ∈ [0,1]×[0,1] such that |Q−R| < δ.
Now divide [0,1] × [0,1] into subsquares of diagonal < δ and pick a
point Q in each subsquare (say, the center). Each subsquare is mapped by
d into the open ball with center p(Q) and radius ε because, if R is in the
same subsquare as Q, wehave|Q − R| < δ and hence |p(Q) − p(R)| < ε.
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Exercises
9.4.1 Show that the function f (x)=1/x is continuous, but not uniformly continuous,
on the open interval (0,1).
9.4.2 Give an example of continuous function that is not uniformly continuous
on GL(2,C).
9.5 Deforming a path in a sequence of small steps
The proof of uniform continuity of path deformations assumes only that d
is a continuous map of the square into R n . We now need to recall how such
a map is interpreted as a “path deformation.” The restriction of d to the
bottom edge of the square is one path p, the restriction to the top edge is
another path q, and the restriction to the various horizontal sections of the
square is a “continuous series” of paths between p and q—a deformation
from p to q. Figure 9.2 shows the “deformation snapshots” of Figure 8.3
further subdivided by vertical sections of the square, thus subdividing the
square into small squares that are mapped to “deformed squares” by d.
−→ d
Im(q)
Im(p)
Figure 9.2: Snapshots of a path deformation.