Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces

¶ This setting extends The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers: its conventions and notation are in force.

¶ A metric space is written $(X,d)$, or $X$ when $d$ is clear. For $x\in X$, the open ball of radius $r>0$ is written $B_{d}(x,r)$ and the closed ball of radius $r\ge0$ is written $\bar{B}_{d}(x,r)$. For a sequence $(x_{n})$ in $X$ that is convergent, its limit is written $\lim_{n\to\infty}x_{n}$.

¶ For a metric space $(X,d)$, $\mathcal{T}_{d}$ denotes the set of all $U\in\mathcal{P}(X)$ that are open in $(X,d)$; it is a topology on $X$, by The Open Subsets of a Metric Space Form a Topology §topology.

¶ $d_{\mathbb{R}}$ is the distance on $\mathbb{R}$, $d_{\mathbb{R}}(s,t)=|s-t|$; it is a metric on $\mathbb{R}$, by The Absolute Value Metric on the Real Numbers §metric, and a map into $\mathbb{R}$ is read as a map into $(\mathbb{R},d_{\mathbb{R}})$. For a sequence in $\mathbb{R}$, convergence, the limit and the Cauchy property in $(\mathbb{R},d_{\mathbb{R}})$ agree with those of Convergent Sequences of Real Numbers §converges, The Limit of a Convergent Sequence §limit and Cauchy Sequences of Real Numbers §cauchy, by Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete §converges, Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete §limit and Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete §cauchy, and $(\mathbb{R},d_{\mathbb{R}})$ is complete, by Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete §complete.

¶ For $z\in\mathbb{C}$, $|z|$ denotes the modulus of $z$, which for real $z$ is its absolute value, by The Modulus of a Complex Number §agreement, and $d_{\mathbb{C}}$ is the distance on $\mathbb{C}$, $d_{\mathbb{C}}(z,w)=|z-w|$; it is a metric on $\mathbb{C}$, by The Modulus Metric on the Complex Numbers §metric, and a map into $\mathbb{C}$ is read as a map into $(\mathbb{C},d_{\mathbb{C}})$. A sequence $(z_{n})$ in $\mathbb{C}$ is convergent in $(\mathbb{C},d_{\mathbb{C}})$ if and only if $(\operatorname{Re}z_{n})$ and $(\operatorname{Im}z_{n})$ are convergent, and then $\lim_{n\to\infty}z_{n}=\bigl(\lim_{n\to\infty}\operatorname{Re}z_{n}\bigr)+\bigl(\lim_{n\to\infty}\operatorname{Im}z_{n}\bigr)i$, by Convergence in the Complex Numbers by Real and Imaginary Parts, and Completeness in the Modulus Metric §components and Convergence in the Complex Numbers by Real and Imaginary Parts, and Completeness in the Modulus Metric §limit, with real and imaginary parts as in Real and Imaginary Parts of a Complex Number §parts; and $(\mathbb{C},d_{\mathbb{C}})$ is complete, by Convergence in the Complex Numbers by Real and Imaginary Parts, and Completeness in the Modulus Metric §complete.

¶ The following are used without further citation. Limits in a metric space are unique, by Uniqueness of Limits in a Metric Space §unique. A subsequence of a sequence that converges to $x$ converges to $x$, by A Subsequence of a Convergent Sequence Has the Same Limit §subsequence. Open balls are open, by Open Ball in a Metric Space is Open §open. A map is continuous at a point relative to a subset, or continuous on it, only if, and under the axiom of dependent choice also if, for every sequence in the subset converging to the point, or to any point of the subset, the image sequence converges to the image of that point, by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset; and a map between metric spaces is continuous if and only if preimages of open sets are open, by Continuity of a Map Between Metric Spaces via Preimages of Open Sets §preimage-open. Lipschitz maps are uniformly continuous and uniformly continuous maps are continuous, by A Lipschitz Map is Uniformly Continuous and Continuous §uniformly-continuous and A Uniformly Continuous Map Between Metric Spaces Is Continuous §continuous. Constant maps into $\mathbb{R}$ are continuous at every point relative to every subset, and pointwise sums, products and real multiples of real-valued maps continuous at a point relative to a subset, or continuous on it, are continuous there, or on it, by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants, Continuity of Sums and Products of Real-Valued Functions on a Metric Space §sums, Continuity of Sums and Products of Real-Valued Functions on a Metric Space §products, Continuity of Sums and Products of Real-Valued Functions on a Metric Space §scalar and Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set.