Topology of Plane Sets of Points 93
is also a metric space, denoted 02 [a, b]. It is called the space of continuous
functions with quadratic metric.
Let (S,d) be any metric space and let M be a nonempty subset of S. If
d M is the restriction of the function d to M, i.e., if d M ( x, y) = d( x, y) for
(x, y) E M x M, it is a trivial matter to verify that dM is a metric in M.
Thus (M, dM) is a metric space, and it is referred to as a subspace of S.
The metric dM is called the relative metric induced on M by d.
EXERCISES 2.3
- Show that the following metrics satisfy the axioms of Definition 2.7.
(a) d(x,y) = Ix - YI on IB.^1
(b) d(z, z^1 ) = lz - z^1 I on C
(c) x(z,z^1 ) on C*
(d) p(z,z^1 ) = Ix - x^1 I + IY - y^1 I on C
(e) p(x,y) = maxlxk - Ykl on IB.n
(f) d(f,g) = maxlf(t)-g(t)I, a:::; t :::;· b, on O[a,b]
- Prove that if d( x, y) is a metric in a set S, then
d(x,y)
p(x,y) = 1 + d(x,y)
i:;i also metric in the same set.
- If, as in problem 2, d(x, y) is a metric in S, prove that
p(x, y) = min[l, d(x, y)]
is also a metric in S.
- Show that the four axioms of Definition 2. 7 can be reduced to the
following two axioms (A. Lindenbaum, 1926):
(a) d( x, y) = 0 iff x = y
(b) d(x,z):::; d(x,y) + d(z,y) for arbitrary x,y,z ES - For any complex number z, let !!zl! = p(z, 0) = lxl + IYI· Prove the
following.
(a) l!z1 + z21! :::; !lz1!1 + !lz2ll
(b) l!z1z2!1 :::; l!z1!1 · l!z21!
(c) !z!2 :::; l!zl!2 :::; 2!z!2
(d) 1/!lz!I :::; 111/zll :::; 2/!!zl! (z -f 0)
2.6 NEIGHBORHOODS. OPEN AND CLOSED SETS
Let (S,d) be any metric space, and let a be any given point in S.
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