Circle as a quotient of the real line

Builds on Quotient topology Universal property of the quotient topology Continuous bijection from a compact space to a Hausdorff space is a homeomorphism

Example

On $\mathbb{R}$ , let $s \sim t$ when $s - t \in \mathbb{Z}$ . Let $\mathbb{R} / \mathbb{Z}$ be the set of classes, with the quotient topology. Let $S^1 = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$ be the unit circle, with the subspace topology of $\mathbb{R}^2$ . Then the map $[t] \mapsto (\cos 2\pi t, \sin 2\pi t)$ is a homeomorphism from $\mathbb{R} / \mathbb{Z}$ to $S^1$ .

Remarks

The map is well defined and continuous by the universal property of the quotient topology, and it is bijective. The space $\mathbb{R} / \mathbb{Z}$ is compact, because it is the image of $[0, 1]$ under the quotient map. The circle is Hausdorff. Thus the map is a homeomorphism, by this theorem. Equivalently, $S^1$ is homeomorphic to $[0, 1]$ with the two endpoints $0$ and $1$ glued together. Addition of classes makes $\mathbb{R} / \mathbb{Z}$ a group, the circle group.

Reading path

To understand this page from the ground up, read these 60 pages in order. The foundations come first.

Element definition
Proposition concept
Real numbers object
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Complement definition
Function definition
Subspace topology definition
Equivalence relation definition
Metric space definition
Preimage of a set definition
Codomain definition
Function composition definition
Injective function definition
Open ball definition
Equivalence class definition
Surjective function definition
Metric topology definition
Quotient set definition
Bijective function definition
Open set definition
Quotient topology definition
Inverse function definition
Hausdorff space definition
Open cover definition
Continuous map of topological spaces definition
Closed set definition
Compact space definition
Complement of an open set is closed statement
Homeomorphism definition
Universal property of the quotient topology statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example

Where this leads

15 pages build on this one.

Tag	`0WH1GVBJNH`
URI	`m2://0WH1GVBJNH`
Aliases	`real-line-modulo-integers`
Kind	example
Writing	W0 stub
MSC	54B15
Instance of	Topological space
Uses	Quotient topology, Universal property of the quotient topology, Continuous bijection from a compact space to a Hausdorff space is a homeomorphism

Circle as a quotient of the real line

Example

Remarks

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