University:
California State University, Northridge
Course:
MATH 501 | Topology
Academic year:

2017
Views:

312

Pages:

2
Author:

Derrick Hunt

EXAMPLES OF INCOMPLETE METRIC SPACES Example 1 Let Q be a the set of all rational numbers with the metric given by d(x, y) = |x − y|, 1 + xn , for 2 + xn 1 + xn ≤ 2, n ≥ 2. The sequence {xn } is an increasing sequence of rational numbers such that xn = 2 2 + xn √ for n ≥ 1. (Can you check it? ) Next, one shows (Can you do it? ) that lim xn = 2 6∈ Q and thus {xn } for x, y ∈ Q. Consider the sequence {xn } of rational numbers such that x1 = 1 and xn+1 = 2 n→∞ is a Cauchy sequence in Q that is not convergent in Q. Thus (Q, d) is not a complete metric space. Example 2 Let X be the set of all continuous real-valued functions on [0, 1] and define a metric on X by d(x, y) = Z1 |x(t) − y(t)| dt, 0 for x, y ∈ X. The metric space (X, d) is not complete. Proof. Define the sequence xm (t) of continuous functions on [0, 1] by   if 0 ≤ t ≤ 21 ; 0, xm (t) = m x − 12 , if 21 < t < am = 12 +  1, if am ≤ t ≤ 1. 1 ; m {xm } is a Cauchy sequence. Indeed, d(xm , xn ) is the area of the shaded triangle in the figure below, 1 m 1 m 1 n 1 1 xn xm xm 1 0 1 2 am 1 2 0 t t 1 1 2 and d(xm , xn ) < ǫ, when n, m > 1/ǫ. Next, we show that xm does not converge in X. For every x ∈ X we have d(xm , x) = Z1 0 Z1/2 Zam Z1 |xm (t) − x(t)| dt = |x(t)| dt + |xm (t) − x(t)| dt + |1 − x(t)| dt. 0 1/2 am The integrands above are nonnegative, so is each integral on the right hand side. Therefore d(xm , x) → 0 as m → ∞ would imply that each integral approaches zero, and, since x(t) is continuous, we should have x(t) = 0 if t ∈ [0, 1/2) and x(t) = 1 if t ∈ (1/2, 1]. But this is impossible for a continuous function. Hence {xm } does not converge. This proves that (X, d) is not a complete metric space.

Examples of Incomplete Metric Spaces

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