Understanding Complete Metric Spaces

A metric space $(X, d)$ is called complete if every Cauchy sequence in $X$ converges to a limit that is also an element of $X$. A sequence $(x_n)$ in a metric space is Cauchy if for every $\epsilon > 0$, there exists a positive integer $N$ such that for all $m, n > N$, the distance between $x_m$ and $x_n$, denoted by $d(x_m, x_n)$, is less than $\epsilon$. Essentially, the terms of a Cauchy sequence get arbitrarily close to each other as the sequence progresses.

Let's examine the given options in the context of complete metric spaces, using the standard metric $|x-y|$ for real and rational numbers, and $|z_1-z_2|$ for complex numbers.

• Set of complex numbers ($\mathbb{C}$): The set of complex numbers $\mathbb{C}$ with the standard metric is a complete metric space. The space $\mathbb{C}$ can be identified with $\mathbb{R}^2$, and the standard metric on $\mathbb{C}$ corresponds to the Euclidean metric on $\mathbb{R}^2$. Since $\mathbb{R}^2$ is known to be complete, $\mathbb{C}$ is also complete.
• Set of rationals ($\mathbb{Q}$): The set of rational numbers $\mathbb{Q}$ with the standard metric is not a complete metric space. We can construct Cauchy sequences of rational numbers whose limits are not rational numbers. For example, consider a sequence of rational numbers $(x_n)$ that converges to $\sqrt{2}$. One such sequence can be generated using Newton's method for finding the roots of $x^2 - 2 = 0$, starting with a rational guess, say $x_1 = 1$. The iteration formula is $x_{n+1} = x_n - \frac{x_n^2 - 2}{2x_n} = \frac{x_n^2 + 2}{2x_n}$. The terms are $x_1=1$, $x_2 = \frac{1^2+2}{2 \cdot 1} = \frac{3}{2}$, $x_3 = \frac{(3/2)^2+2}{2 \cdot 3/2} = \frac{9/4+2}{3} = \frac{17/4}{3} = \frac{17}{12}$, and so on. All these terms are rational numbers. This sequence is a Cauchy sequence in $\mathbb{Q}$, but it converges to $\sqrt{2}$, which is an irrational number. Since $\sqrt{2} \notin \mathbb{Q}$, this Cauchy sequence does not converge within $\mathbb{Q}$. Therefore, $\mathbb{Q}$ is not complete.
• Set of irrationals ($\mathbb{R} \setminus \mathbb{Q}$): The set of irrational numbers $\mathbb{R} \setminus \mathbb{Q}$ with the standard metric is also not a complete metric space. For example, consider a sequence of irrational numbers $(y_n)$ converging to 1, such as $y_n = 1 + \frac{\sqrt{2}}{n}$. As $n \to \infty$, $y_n \to 1$. Each $y_n$ is irrational, but the limit 1 is rational. This sequence is Cauchy in $\mathbb{R} \setminus \mathbb{Q}$, but its limit is not in $\mathbb{R} \setminus \mathbb{Q}$. Therefore, $\mathbb{R} \setminus \mathbb{Q}$ is not complete.
• Set of real numbers ($\mathbb{R}$): The set of real numbers $\mathbb{R}$ with the standard metric is a complete metric space. This is a fundamental property of the real number system, often established through its construction (e.g., via Dedekind cuts or as the completion of the rational numbers). Every Cauchy sequence of real numbers converges to a unique real number.

Comparing the options, the set of rationals and the set of irrationals are not complete metric spaces. The question asks for one metric space that is not complete. Based on our analysis, the set of rationals is one such space.

Revision Table: Complete vs. Incomplete Metric Spaces

Additional Information: Understanding Completion

For any metric space that is not complete, it is possible to construct its completion. The completion of a metric space $(X, d)$ is a complete metric space $(\bar{X}, \bar{d})$ that contains $X$ as a dense subspace. The most famous example is the completion of the set of rational numbers $\mathbb{Q}$. The completion of $\mathbb{Q}$ is the set of real numbers $\mathbb{R}$. This means that $\mathbb{R}$ is the smallest complete metric space containing $\mathbb{Q}$.

Understanding completeness is crucial in analysis and topology, especially when dealing with convergence of sequences and series, continuity of functions, and the existence of limits.