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Dixmier always solves this as follows, e.g. in C*-algebras — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book form which to learn about C*-algebras”):

16.1. The compact group associated with a topological group

16.1.1. Theorem. Let $G$ be a topological group. There exists a compact group $\Sigma$ and a continuous morphism $\alpha:G\to\Sigma$ possessing the following property: (...). Furthermore, the pair $(\Sigma,\alpha)$ is determined up to isomorphism by this property.

(... Long proof goes here ...)

16.1.2. Definition. The group $\Sigma$ is called the compact group associated with $G$, and $\alpha$ is called the canonical morphism of $G$ into $\Sigma$.

The book has about two dozen “X.Y.2” definitions like this — e.g. no less than six over pp. 116–123 (disjoint representations, factor representation, quasi-equivalent representations, type I, multiplicity-free, multiplicity), and the almost last statement in the book is a definition (Plancherel measure).

Added: I took the above example as perhaps the closest to yours (constructed object). In fact Dixmier’s relative Bourbaki does yours exactly, just the same down to the use of a subtitle (Algebra I.2.4 “Monoid of fractions”, I.8.12 “Rings of fractions”, I.9.4 “The field of rational numbers”):

12. Rings of fractions

Theorem 4. (...)

Definition 8. The ring defined in Theorem 4 is called the ring of fractions (...)

Of course, this “classic” way is not the only one: I also agree with Nik Weaver’s answer, and with this contrasting epigraph in Reed & Simon, Functional Analysis (my emphasis): “A good definition should be the hypothesis of a theorem. (J. Glimm)”

Dixmier always solves this as follows, e.g. in C*-algebras — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book from which to learn about C*-algebras”):

16.1. The compact group associated with a topological group

16.1.1. Theorem. Let $G$ be a topological group. There exists a compact group $\Sigma$ and a continuous morphism $\alpha:G\to\Sigma$ possessing the following property: (...). Furthermore, the pair $(\Sigma,\alpha)$ is determined up to isomorphism by this property.

(... Long proof goes here ...)

16.1.2. Definition. The group $\Sigma$ is called the compact group associated with $G$, and $\alpha$ is called the canonical morphism of $G$ into $\Sigma$.

The book has about two dozen “X.Y.2” definitions like this — e.g. no less than six over pp. 116–123 (disjoint representations, factor representation, quasi-equivalent representations, type I, multiplicity-free, multiplicity), and the almost last statement in the book is a definition (Plancherel measure).

Added: I took the above example as perhaps the closest to yours (constructed object). In fact Dixmier’s relative Bourbaki does yours exactly, just the same down to the use of a subtitle (Algebra I.2.4 “Monoid of fractions”, I.8.12 “Rings of fractions”, I.9.4 “The field of rational numbers”):

12. Rings of fractions

Theorem 4. (...)

Definition 8. The ring defined in Theorem 4 is called the ring of fractions (...)

Of course, this “classic” way is not the only one: I also agree with Nik Weaver’s answer, and with this contrasting epigraph in Reed & Simon, Functional Analysis (my emphasis): “A good definition should be the hypothesis of a theorem. (J. Glimm)”

Dixmier always solves this as follows, e.g. in C*-algebras — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book form which to learn about C*-algebras”):

16.1. The compact group associated with a topological group

16.1.1. Theorem. Let $G$ be a topological group. There exists a compact group $\Sigma$ and a continuous morphism $\alpha:G\to\Sigma$ possessing the following property: (...). Furthermore, the pair $(\Sigma,\alpha)$ is determined up to isomorphism by this property.

(... Long proof goes here ...)

16.1.2. Definition. The group $\Sigma$ is called the compact group associated with $G$, and $\alpha$ is called the canonical morphism of $G$ into $\Sigma$.

The book has about two dozen “X.Y.2” definitions like this — e.g. no less than six over pp. 116–123 (disjoint representations, factor representation, quasi-equivalent representations, type I, multiplicity-free, multiplicity), and the almost last statement in the book is a definition (Plancherel measure).

Added: I took the above example as perhaps the closest to yours (constructed object). In fact Dixmier’s relative Bourbaki does yours exactly, just the same down to the use of a subtitle (Algebra I.2.4 “Monoid of fractions”, I.8.12 “Rings of fractions”, I.9.4 “The field of rational numbers”):

12. Rings of fractions

Theorem 4. (...)

Definition 8. The ring defined in Theorem 4 is called the ring of fractions (...)

Of course, this “classic” way is not the only one: I also agree with Nik Weaver’s answer, and with this contrasting epigraph in Reed & Simon, Functional Analysis: “A good definition should be the hypothesis of a theorem. (J. Glimm)”

Dixmier always solves this as follows, e.g. in C*-algebras — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book form which to learn about C*-algebras”):

16.1. The compact group associated with a topological group

16.1.1. Theorem. Let $G$ be a topological group. There exists a compact group $\Sigma$ and a continuous morphism $\alpha:G\to\Sigma$ possessing the following property: (...). Furthermore, the pair $(\Sigma,\alpha)$ is determined up to isomorphism by this property.

(... Long proof goes here ...)

16.1.2. Definition. The group $\Sigma$ is called the compact group associated with $G$, and $\alpha$ is called the canonical morphism of $G$ into $\Sigma$.

The book has about two dozen “X.Y.2” definitions like this — e.g. no less than six over pp. 116–123 (disjoint representations, factor representation, quasi-equivalent representations, type I, multiplicity-free, multiplicity), and the almost last statement in the book is a definition (Plancherel measure).

Added: I took the above example as perhaps the closest to yours (constructed object). In fact Dixmier’s relative Bourbaki does yours exactly, just the same down to the use of a subtitle (Algebra I.2.4 “Monoid of fractions”, I.8.12 “Rings of fractions”, I.9.4 “The field of rational numbers”):

12. Rings of fractions

Theorem 4. (...)

Definition 8. The ring defined in Theorem 4 is called the ring of fractions (...)

Of course, this “classic” way is not the only one: I also agree with Nik Weaver’s answer, and with this contrasting epigraph in Reed & Simon, Functional Analysis (my emphasis): “A good definition should be the hypothesis of a theorem. (J. Glimm)”

Dixmier always solves this as follows, e.g. in C*-algebras — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book form which to learn about C*-algebras”):

16.1. The compact group associated with a topological group

16.1.1. Theorem. Let $G$ be a topological group. There exists a compact group $\Sigma$ and a continuous morphism $\alpha:G\to\Sigma$ possessing the following property: for every compact group $\Sigma'$ and every continuous morphism $\alpha':G\to\Sigma'$, there exists a unique continuous morphism $\beta:\Sigma\to\Sigma'$ such that $\alpha'= \beta\circ\alpha$. Furthermore, the pair $(\Sigma,\alpha)$ is determined up to isomorphism by this property.

(... Long proof goes here ...)

16.1.2. Definition. The group $\Sigma$ is called the compact group associated with $G$, and $\alpha$ is called the canonical morphism of $G$ into $\Sigma$.

The book has about two dozen “X.Y.2” definitions like this — e.g. no less than six over pp. 116–123 (disjoint representations, factor representation, quasi-equivalent representations, type I, multiplicity-free, multiplicity), and the almost last statement in the book is a definition (Plancherel measure). Of course, this is not the only way — contrast an epigraph in Reed & Simon, Functional Analysis: “A good definition should be the hypothesis of a theorem. (J. Glimm)”

Added: I took the above example (constructed object) as perhaps the closest to yours. In fact, I should have gone to Dixmier’s relative Bourbaki for it exactly (Algebra I.2.4 “Monoid of fractions”, I.8.12 “Rings of fractions”, I.9.4 “The field of rational numbers”), and stressed their use of subtitles:

12. Rings of fractions

Theorem 4. (...)

Definition 8. The ring defined in Theorem 4 is called the ring of fractions (...)

Dixmier always solves this as follows, e.g. in C*-algebras — surely one possible example of good exposition (E. C. Lance’s translator‘s preface: “With is clear and straightforward style, this remains the best book form which to learn about C*-algebras”):

16.1. The compact group associated with a topological group

16.1.1. Theorem. Let $G$ be a topological group. There exists a compact group $\Sigma$ and a continuous morphism $\alpha:G\to\Sigma$ possessing the following property: (...). Furthermore, the pair $(\Sigma,\alpha)$ is determined up to isomorphism by this property.

(... Long proof goes here ...)

16.1.2. Definition. The group $\Sigma$ is called the compact group associated with $G$, and $\alpha$ is called the canonical morphism of $G$ into $\Sigma$.

The book has about two dozen “X.Y.2” definitions like this — e.g. no less than six over pp. 116–123 (disjoint representations, factor representation, quasi-equivalent representations, type I, multiplicity-free, multiplicity), and the almost last statement in the book is a definition (Plancherel measure).

Added: I took the above example as perhaps the closest to yours (constructed object). In fact Dixmier’s relative Bourbaki does yours exactly, just the same down to the use of a subtitle (Algebra I.2.4 “Monoid of fractions”, I.8.12 “Rings of fractions”, I.9.4 “The field of rational numbers”):

12. Rings of fractions

Theorem 4. (...)

Definition 8. The ring defined in Theorem 4 is called the ring of fractions (...)

Of course, this “classic” way is not the only one: I also agree with Nik Weaver’s answer, and with this contrasting epigraph in Reed & Simon, Functional Analysis: “A good definition should be the hypothesis of a theorem. (J. Glimm)”