Answer Happy

1.7+) (-, f(x)+p) x+p. 32 LA - an (3,A(=)) (5,2)-P) (1,0) ام-کم (0,0) Figure 2.6 Figure 2.5 D., Ds, and D, of Examples 2 and 3. The reader should be sure to verify these figures Note that the D.-I-neighborhood of (0,0) is a subset of the D-l. neighborhood of (0,0). Since D((21,41), (229)) < D.((+1, 1), (, va)) for any two points (31,9) and (22,W2) of P* (Exercise 2), the Di-p- neighborhood of any point of R is a subset of the D-p-neighbor- hood for any positive real number p. However, a simple calculation shows that the D-p/V2-neighborhood of (t', y) is a subset of the Di-p neighborhood of (2) (Fig. 2.5). Note, however, that if p 31, there is no positive number such that either the D-g-neighborhood of .) or the D1-9-neighborhood of (',) is a subset of the D2-p-neigh- borhood of (*',V) which consists of (+',') alone. Example 7. Let X, D be the metric space described in Example 5. Suppose € X and p > 0. We may draw a p-collar about the graph off as shown in Fig. 2.6. Then the D-p-neighborhood off will consist of all functions from (0, 1) into [0, 1] whose graphs lie within the p-collar of f. EXERCISES 1. Confirm Figs. 2.1 through 2.4. 2. Prove that D((21. vi), (23, va)) < D:((21, 1), (22, ya)) for any two points (21. y) and (3, va) in Rº as claimed in Example 6. Also in Example 6, carry out the computation which shows that the D-p/V2-neighborhood of (x,y) is a subset of the D1-p-neighborhood of (z,) for any (z,y) E R.
2.3 Open Soto 21 3. Let (x, y) be any point of Ra, and p be any positive number. What is the relation of the D3-p-neighborhood of (x,y) to the D-p-neighborhood of (7, y)? to the D1-p-neighborhood of (x, y)? Prove that every D- and Di- neighborhood of (x, y) contains a D3-neighborhood of (x, y) and that each D3-neighborhood of (x, y) contains both a D- and a Dı-neighborhood. 4. Let X, D be the metric space as described in Example 5. Sketch the graphs of the elements of X defined by each of the following, and sketch the p-collars (Example 6) for each of these elements. a) f(t) = 1, for all 2 € (0, 1). b) f(x) = x, for all 2 € (0, 1). c) f(x) = 73, for all z € (0, 1). d) f(x) (1, if 0 < r < 5; 12, if : << 1.
Example 5. This example is given to illustrate that a metric can be defined on a set which is neither R" for some n nor a subset of R". Let X be the set of all functions from the closed interval (0, 1) into itself. If f and g are any such functions, define D(, 9) = - least upper bound {\f(x) - g(x)|| 2 € (0, 11). Since any subset of the real numbers which has an upper bound has a least upper bound and 0 < \f() - g(x) < 1 for all f,gEX and IE (0, 1), then D(5,9) is defined for all f, g EX. We will now show that D is a metric for X by showing that D satisfies each of the properties required for a metric in Definition 1: i) DC, 9) 2 0 for all f, 9 € X. Since each element of {\f(x) — 9(2)||* € [0, 1]} greater than or equal to 0, the least upper bound of this set, D(5,9), greater than or equal to 0. ii) DC, 9) = D(9,8) for all f, ge X. This follows at once from the fact that \f(x) - 9(2)) = 19() - f(x). ii) D, 9) = 0 if and only if f= g. If f= g, then f(x) = g(x) for all of 10, 11, and hence (IS(2) - (2)||& € (0,11} = {0}. Therefore it follows that D, 9) = 0. On the other hand, if Df, 9) = 0, then lub {]f(x) - g(x)||* = [0, 1]} = 0. is is