Circle

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Definition

As a subset of the Euclidean plane

A circle with center $O$ and radius $r>0$ is defined as the set of all points $P$ in the Euclidean plane at a distance of $r$ from $O$.

The unit circle is the circle whose center is at the origin and radius is $1$, it is defined as the following subset of the Euclidean plane:

$\{(x,y)\mid x^2+y^2=1\}$

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 $1$.

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:

1. It is the quotient $\mathbb{R}/\mathbb{Z}$ 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.
2. It is the quotient of the closed unit interval $[0,1]$ by the identification of the two points $0$ and $1$. In symbols, this is $[0,1]/\{0,1\}$.

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.

Equivalent spaces

Space	How strongly is it equivalent to the circle (in the geometric sense)?
Ellipse in ${\mathbb{R}}^2$	Equivalent up to an affine transformation
Simple closed convex curve of ${\mathbb{R}}^2$	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$ arising from a straight line homotopy
Simple closed curve in ${\mathbb{R}}^2$	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$
Compact differential 1-manifold	Diffeomorphic
Compact 1-manifold	Homeomorphic
Boundary of 2-simplex (i.e., triangle)	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$ arising from a straight line homotopy
Boundary of a 2-cube (i.e., square)	Equivalent up to a self-homeomorphism of ${\mathbb{R}}^2$ arising from a straight line homotopy

Algebraic topology

Homology groups

Further information: homology computation for spheres

With coefficients in $\mathbb{Z}$, we have $H_0(S^1)\cong H_1(S^1)\cong \mathbb{Z}$. All the higher homology groups are zero.

For reduced homology groups, we have ${\tilde{H}}_1(S^1)\cong \mathbb{Z}$ and all the other reduced homology groups are zero.

More generally, for any module $M$ over any commutative unital ring $R$, we have $H_0(S^1;M)\cong H_1(S^1;M)\cong M$ and all higher homology groups are zero. For reduced homology groups, ${\tilde{H}}_1(S^1;M)\cong M$ and all the other reduced homology groups are zero.

Cohomology groups

With coefficients in $\mathbb{Z}$, we have $H^0(S^1)\cong H^1(S^1)\cong \mathbb{Z}$, and all the higher homology groups are zero. The cohomology ring is isomorphic to $\mathbb{Z}[x]/(x^2)$ where $x$ is an additive generator of the first cohomology group.

More generally, with coefficients in any commutative unital ring $R$, we have $H^0(S^1;R)\cong H^1(S^1;R)\cong R$ and the cohomology ring is isomorphic to $R[x]/(x^2)$ where $x$ is an additive generator of the first cohomology group.

Homotopy groups

Further information: homotopy computation for spheres

We have ${\pi }_0(S^1)$ is the one-point set (or trivial group, if we use the H-space interpretation of $S^1$). The fundamental group ${\pi }_1(S^1)$ is isomorphic to the group of integers $\mathbb{Z}$, and all the higher homotopy groups are trivial.

In particular, the circle is a realization of the classifying space for the group of integers $\mathbb{Z}$. Its universal cover, the Euclidean line, is a contractible space.