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As a subset of the Euclidean plane
A circle with center and radius is defined as the set of all points in the Euclidean plane at a distance of from .
The unit circle is the circle whose center is at the origin and radius is , it is defined as the following subset of the Euclidean plane:
Under the identification of the Euclidean plane with the complex numbers, this can also be described as the set of complex numbers whose modulus is .
Note that all circles are equivalent up to similarity transformations of the Euclidean plane.
As a quotient
The circle is defined as a quotient in either of the following equivalent ways:
- It is the quotient in the sense of topological spaces. In other words it is the quotient of real numbers by the equivalence relation of differing by an integer.
- It is the quotient of the closed unit interval by the identification of the two points and . In symbols, this is .
As a topological space
As a topological space, a circle is defined as the topological space obtained by using the subspace topology from the Euclidean plane on any circle described above.
|Space||How strongly is it equivalent to the circle (in the geometric sense)?|
|Ellipse in||Equivalent up to an affine transformation|
|Simple closed convex curve of||Equivalent up to a self-homeomorphism of arising from a straight line homotopy|
|Simple closed curve in||Equivalent up to a self-homeomorphism of|
|Compact differential 1-manifold||Diffeomorphic|
|Boundary of 2-simplex (i.e., triangle)||Equivalent up to a self-homeomorphism of arising from a straight line homotopy|
|Boundary of a 2-cube (i.e., square)||Equivalent up to a self-homeomorphism of arising from a straight line homotopy|
Further information: homology computation for spheres
With coefficients in , we have . All the higher homology groups are zero.
For reduced homology groups, we have and all the other reduced homology groups are zero.
More generally, for any module over any commutative unital ring , we have and all higher homology groups are zero. For reduced homology groups, and all the other reduced homology groups are zero.
With coefficients in , we have , and all the higher homology groups are zero. The cohomology ring is isomorphic to where is an additive generator of the first cohomology group.
More generally, with coefficients in any commutative unital ring , we have and the cohomology ring is isomorphic to where is an additive generator of the first cohomology group.
Further information: homotopy computation for spheres
We have is the one-point set (or trivial group, if we use the H-space interpretation of ). The fundamental group is isomorphic to the group of integers , and all the higher homotopy groups are trivial.
Algebraic and coalgebraic structure
The circle has the natural structure of an abelian group, which can be realized in many ways:
- View it as the set of complex numbers with modulus 1, and perform multiplication of complex numbers.
- View it as , i.e., the quotient of the additive group of real numbers by the discrete subgroup of integers.
Also, on account of being a topological group, the circle gets a H-space structure. Thus, its fundamental group is an abelian group and its zeroth homotopy set (which is a one-point set) naturally acquires a group structure (giving the trivial group).
Further information: comultiplication of spheres
Fix a base point . We can construct a comultiplication:
where denotes the wedge sum and the map is a continuous based map, i.e., a continuous map preserving basepoint. This map is cocommutative and coassociative up to homotopy, and it is used to give a group structure to the set of homotopy classes from the based circle to any based topological space. This group is termed the fundamental group.