Metric Spaces: Topologies, Completeness & Baire's Category Theorem

1. The Metric Axioms

The concept of a metric space, introduced by Maurice Fréchet in 1906, abstracts the intuitive notion of distance between two points into a rigorous mathematical framework.

Definition 1.1 (Metric Space): Let $X$ be a non-empty set. A metric (or distance function) on $X$ is a function $d: X \times X \to \mathbb{R}$ satisfying the following four axioms for all $x, y, z \in X$:

1. Non-negativity: $d(x, y) \ge 0$.

2. Identity of Indiscernibles: $d(x, y) = 0 \iff x = y$.

3. Symmetry: $d(x, y) = d(y, x)$.

4. Triangle Inequality: $d(x, z) \le d(x, y) + d(y, z)$.

The ordered pair $(X, d)$ is called a metric space. The elements of $X$ are called points.

2. Classical Examples of Metric Spaces

Example 1.1 (Euclidean and $\ell^p$ Spaces on $\mathbb{R}^n$): For $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ in $\mathbb{R}^n$:

• Euclidean Metric ($p = 2$):

• Taxicab / Manhattan Metric ($p = 1$):

• Chebyshev / Maximum Metric ($p = \infty$):

• General $\ell^p$ Metric ($1 \le p < \infty$):

The triangle inequality for $d_p$ is precisely Minkowski's Inequality:

Example 1.2 (Sequence Space $\ell^\infty$ and $\ell^p$): Let $\ell^\infty$ denote the space of all bounded real sequences $x = (x_n)_{n=1}^\infty$ with norm $\|x\|_\infty = \sup_{n \ge 1} |x_n|$. The metric is:

For $1 \le p < \infty$, the space $\ell^p$ consists of sequences with $\sum_{n=1}^\infty |x_n|^p < \infty$, with metric:

Example 1.3 (Function Space $C[a, b]$ with Uniform Metric): Let $C[a, b]$ be the set of all real-valued continuous functions on the closed interval $[a, b]$. The uniform metric (or Chebyshev metric) is:

By the Extreme Value Theorem, the maximum is always attained and finite.

Example 1.4 (The Discrete Metric): For any non-empty set $X$, the discrete metric is defined by:

It satisfies all metric axioms trivially.

Example 1.5 (Ultrametric Spaces): A metric space $(X, d)$ is an ultrametric space if it satisfies the strong triangle inequality:

A canonical example is the $p$-adic metric $d_p(x, y) = |x - y|_p$ on $\mathbb{Q}$. In an ultrametric space, every triangle is isosceles, and every point inside an open ball is its center!

1. Open and Closed Balls

Definition 1.2 (Open and Closed Balls): Let $(X, d)$ be a metric space, $x_0 \in X$, and $r > 0$.

• The open ball of radius $r$ centered at $x_0$ is:

• The closed ball of radius $r$ centered at $x_0$ is:

• The sphere of radius $r$ centered at $x_0$ is:

2. Open Sets and the Metric Topology

Definition 1.3 (Open Set in a Metric Space): A subset $U \subseteq X$ is called open in $(X, d)$ if for every point $x \in U$, there exists a radius $r > 0$ such that the entire open ball is contained in $U$:

Theorem 1.1 (Properties of Open Sets): Let $(X, d)$ be a metric space.

1. The empty set $\emptyset$ and the whole space $X$ are open.

2. The union of any arbitrary family of open sets is open:

3. The intersection of any finite collection of open sets is open:

Proof:

1. $\emptyset$ is open vacuously (there are no points in $\emptyset$). For $X$, for any $x \in X$, $B(x, 1) \subseteq X$ holds trivially.

2. Let $U = \bigcup_{\alpha \in I} U_\alpha$. For any $x \in U$, $x \in U_{\alpha_0}$ for some $\alpha_0 \in I$. Since $U_{\alpha_0}$ is open, there exists $r > 0$ such that $B(x, r) \subseteq U_{\alpha_0} \subseteq U$. Thus $U$ is open.

3. Let $V = \bigcap_{i=1}^n U_i$. For any $x \in V$, $x \in U_i$ for all $i \in \{1, \dots, n\}$.

Since each $U_i$ is open, there exists $r_i > 0$ such that $B(x, r_i) \subseteq U_i$. Let $r = \min\{r_1, r_2, \dots, r_n\} > 0$ (since $n$ is finite!). Then for each $i$, $B(x, r) \subseteq B(x, r_i) \subseteq U_i$. Hence $B(x, r) \subseteq \bigcap_{i=1}^n U_i = V$, proving $V$ is open. $\blacksquare$

Definition 1.4 (Topology Induced by a Metric): The collection $\mathcal{T}_d = \{U \subseteq X : U \text{ is open in } (X, d)\}$ is called the topology induced by the metric $d$.

3. Equivalent Metrics

Two distinct metrics on the same underlying set $X$ can generate the exact same family of open sets.

Definition 1.5 (Topological and Lipschitz Equivalence): Let $d_1$ and $d_2$ be two metrics on $X$.

1. $d_1$ and $d_2$ are topologically equivalent if they induce the same topology: $\mathcal{T}_{d_1} = \mathcal{T}_{d_2}$.

2. $d_1$ and $d_2$ are strongly (Lipschitz) equivalent if there exist constants $c_1, c_2 > 0$ such that for all $x, y \in X$:

Proposition 1.1: Lipschitz equivalence implies topological equivalence. In particular, on $\mathbb{R}^n$, the metrics $d_1, d_2, d_\infty$ and all $d_p$ ($1 \le p < \infty$) are strongly equivalent:

and therefore they all generate the identical Euclidean topology on $\mathbb{R}^n$.

1. Convergence of Sequences

Definition 1.6 (Sequence Convergence): A sequence $(x_n)_{n=1}^\infty$ in a metric space $(X, d)$ is said to converge to a limit $x \in X$ (denoted $x_n \to x$ or $\lim_{n \to \infty} x_n = x$) if:

Proposition 1.2 (Uniqueness of Limits): In any metric space $(X, d)$, limits of sequences are unique: If $x_n \to x$ and $x_n \to y$, then $x = y$.

Proof: By the triangle inequality:

For any $\varepsilon > 0$, choose $N$ large enough that $d(x_n, x) < \varepsilon/2$ and $d(x_n, y) < \varepsilon/2$ for all $n \ge N$. Then $d(x, y) < \varepsilon/2 + \varepsilon/2 = \varepsilon$. Since this holds for all $\varepsilon > 0$, we have $d(x, y) = 0 \implies x = y$. $\blacksquare$

2. Cauchy Sequences and Completeness

Definition 1.7 (Cauchy Sequence & Complete Metric Space):

1. A sequence $(x_n)_{n=1}^\infty$ in $(X, d)$ is a Cauchy sequence if:

2. A metric space $(X, d)$ is called complete if every Cauchy sequence in $X$ converges to a limit point that belongs to $X$.

Every convergent sequence is Cauchy, but the converse is not always true!

• $(\mathbb{R}, |\cdot|)$ is complete.

• $(\mathbb{Q}, |\cdot|)$ is not complete (e.g. sequence of rational approximations to $\sqrt{2}$ is Cauchy in $\mathbb{Q}$, but its limit $\sqrt{2} \notin \mathbb{Q}$).

• The open interval $(0, 1)$ with the standard metric is not complete (the sequence $x_n = 1/n$ is Cauchy, but its limit $0 \notin (0, 1)$).

3. Cantor's Intersection Theorem

How is completeness characterized purely in terms of closed sets?

Definition 1.8 (Diameter of a Subset): For non-empty $A \subseteq X$, the diameter is $\operatorname{diam}(A) = \sup \{d(x, y) : x, y \in A\}$.

Theorem 1.2 (Cantor's Intersection Theorem): A metric space $(X, d)$ is complete if and only if for every nested sequence of non-empty closed subsets:

satisfying $\lim_{n \to \infty} \operatorname{diam}(F_n) = 0$, the intersection contains exactly one point:

Proof: ($\Rightarrow$) Assume $X$ is complete: For each $n \in \mathbb{N}$, choose $x_n \in F_n$. Since $F_m \subseteq F_n$ for all $m \ge n$, both $x_m, x_n \in F_n$. Thus $d(x_m, x_n) \le \operatorname{diam}(F_n)$. Since $\lim_{n \to \infty} \operatorname{diam}(F_n) = 0$, given $\varepsilon > 0$, choose $N$ such that $\operatorname{diam}(F_N) < \varepsilon$. Then for all $m, n \ge N$, $d(x_m, x_n) < \varepsilon$. Hence $(x_n)$ is a Cauchy sequence! Because $X$ is complete, there exists $x^ \in X$ such that $x_n \to x^$. Since each $F_k$ is closed and the tail $(x_n)_{n=k}^\infty \subseteq F_k$, the limit $x^ \in F_k$ for every $k$. Thus $x^ \in \bigcap_{n=1}^\infty F_n$. If $y \in \bigcap F_n$, then $d(x^, y) \le \operatorname{diam}(F_n) \to 0$, forcing $y = x^$.

($\Leftarrow$) Assume the nested intersection property holds: Let $(x_n)$ be any Cauchy sequence in $X$. For each $k \in \mathbb{N}$, let $A_k = \{x_n : n \ge k\}$ and let $F_k = \overline{A_k}$ be its closure. Then $F_1 \supseteq F_2 \supseteq \cdots$ is a nested sequence of non-empty closed sets. Since $(x_n)$ is Cauchy, $\operatorname{diam}(F_k) = \operatorname{diam}(A_k) \to 0$. By hypothesis, there is a unique $x^ \in \bigcap F_k$. Since $x^ \in \overline{A_k}$, $d(x_n, x^) \le \operatorname{diam}(F_k) \to 0$, so $x_n \to x^ \in X$. Hence $X$ is complete. $\blacksquare$

1. Topological Meagerness (First and Second Category)

René Baire introduced the notions of "category" in 1899 to measure the topological size and density of sets.

Definition 1.9 (Dense and Nowhere Dense Sets): Let $(X, d)$ be a metric space.

1. A subset $A \subseteq X$ is dense in $X$ if its closure is the entire space: $\bar{A} = X$.

2. A subset $E \subseteq X$ is nowhere dense if its closure has empty interior:

Equivalently, every non-empty open set $U$ contains a non-empty open ball disjoint from $E$.

Definition 1.10 (Meager / First Category Sets):

1. A subset $M \subseteq X$ is called meager (or of first category) in $X$ if it is a countable union of nowhere dense sets:

2. A subset that is not meager is called of second category (or non-meager).

3. A subset $R \subseteq X$ is called residual (or comeager) if its complement $X \setminus R$ is meager.

2. Baire's Category Theorem

Baire's Theorem asserts that complete metric spaces cannot be topologically small!

Theorem 1.3 (Baire's Category Theorem - BCT): Let $(X, d)$ be a complete metric space.

1. If $\{U_n\}_{n=1}^\infty$ is a countable collection of dense open subsets of $X$, then their intersection is dense in $X$:

2. In particular, a complete metric space $X \ne \emptyset$ is of second category in itself; that is, $X$ cannot be represented as a countable union of nowhere dense sets.

Proof: We prove formulation (1). Let $W \subseteq X$ be any non-empty open set. We must prove that $W \cap \left(\bigcap_{n=1}^\infty U_n\right) \ne \emptyset$.

Step 1: Since $U_1$ is dense in $X$, $W \cap U_1$ is a non-empty open set. Therefore, we can choose a point $x_1 \in W \cap U_1$ and a radius $0 < r_1 < 1$ such that the closed ball satisfies:

Step 2: Now consider $B(x_1, r_1)$ and $U_2$. Since $U_2$ is dense in $X$, the open set $B(x_1, r_1) \cap U_2$ is non-empty. Hence, we can choose a point $x_2$ and radius $0 < r_2 < r_1 / 2 < 1/2$ such that:

Inductive Step: Continuing this process inductively: for each $n \ge 2$, having constructed $\bar{B}(x_{n-1}, r_{n-1})$, since $U_n$ is dense and $B(x_{n-1}, r_{n-1})$ is non-empty and open, their intersection is non-empty. We choose $x_n \in B(x_{n-1}, r_{n-1}) \cap U_n$ and $0 < r_n < r_{n-1}/2 < 2^{-n}$ such that:

We have constructed a nested sequence of non-empty closed balls:

with $\operatorname{diam}(\bar{B}(x_n, r_n)) \le 2 r_n < 2^{1-n} \to 0$ as $n \to \infty$.

Conclusion via Cantor's Intersection Theorem: Since $(X, d)$ is complete, by Cantor's Intersection Theorem (Theorem 1.2), there exists a point $x^* \in X$ such that:

By construction:

• For $n = 1$: $x^ \in \bar{B}(x_1, r_1) \subseteq W \cap U_1 \implies x^ \in W$.

• For every $n \ge 1$: $x^ \in \bar{B}(x_n, r_n) \subseteq U_n \implies x^ \in U_n$.

Therefore, $x^* \in W \cap \left( \bigcap_{n=1}^\infty U_n \right)$. Since $W$ was an arbitrary non-empty open set, the intersection $\bigcap_{n=1}^\infty U_n$ intersects every non-empty open set in $X$. Thus, $\bigcap_{n=1}^\infty U_n$ is dense in $X$. $\blacksquare$

1. Completeness of $C[a, b]$ Under the Uniform Metric

Theorem 1.4 (Completeness of $(C[a, b], d_\infty)$): The metric space of continuous functions $(C[a, b], d_\infty)$ equipped with the uniform metric $d_\infty(f, g) = \max_{t \in [a, b]} |f(t) - g(t)|$ is a complete metric space (a Banach space).

Proof: Let $(f_n)_{n=1}^\infty$ be a Cauchy sequence in $(C[a, b], d_\infty)$. Given $\varepsilon > 0$, there exists $N \in \mathbb{N}$ such that for all $m, n \ge N$:

For each fixed point $t_0 \in [a, b]$:

Thus, the real sequence $(f_n(t_0))_{n=1}^\infty$ is Cauchy in $\mathbb{R}$. Since $\mathbb{R}$ is complete, it converges to a real number. Define the pointwise limit function:

Letting $m \to \infty$ in $|f_n(t) - f_m(t)| < \varepsilon$, we obtain:

This means $f_n \to f$ uniformly on $[a, b]$. By the Uniform Limit Theorem of analysis, the uniform limit of continuous functions is continuous. Thus $f \in C[a, b]$, and $d_\infty(f_n, f) \to 0$. Hence $(C[a, b], d_\infty)$ is complete. $\blacksquare$

2. The Banach Fixed-Point Contraction Mapping Theorem

One of the most powerful analytical applications of complete metric spaces is Stefan Banach's 1922 Fixed-Point Theorem.

Definition 1.11 (Contraction Mapping): Let $(X, d)$ be a metric space. A mapping $T: X \to X$ is called a contraction if there exists a constant $0 \le k < 1$ (the contraction factor) such that:

Theorem 1.5 (Banach Fixed-Point Theorem): Let $(X, d)$ be a non-empty complete metric space, and let $T: X \to X$ be a contraction mapping with factor $k \in [0, 1)$. Then:

1. $T$ has a unique fixed point $x^ \in X$ (i.e., $T(x^) = x^*$).

2. For any arbitrary starting point $x_0 \in X$, the Picard iteration sequence defined by $x_{n+1} = T(x_n)$ converges to $x^*$:

3. The error estimate satisfies:

Proof: Existence: For any $n \in \mathbb{N}$, by induction on the contraction property:

For any $m > n$:

Since $0 \le k < 1$, $\lim_{n \to \infty} k^n = 0$. Thus, for any $\varepsilon > 0$, choosing $N$ large enough makes $d(x_n, x_m) < \varepsilon$ for all $m > n \ge N$. Hence $(x_n)$ is a Cauchy sequence in $X$.

Because $(X, d)$ is complete, there exists $x^ \in X$ such that $x_n \to x^$. Since contractions are Lipschitz continuous ($d(T(x), T(y)) \le k d(x, y)$):

Thus $x^*$ is indeed a fixed point.

Uniqueness: If $y^ \in X$ is another fixed point ($T(y^) = y^*$), then:

Since $1 - k > 0$ and $d(x^, y^) \ge 0$, we must have $d(x^, y^) = 0 \implies x^ = y^$. $\blacksquare$