Examples of a cone in $\mathbb{R}^2$

Question

$X \subset \mathbb{R}^2$ is a cone if for any $x\in X,\lambda \in \mathbb{R^{>0}}$, then $\lambda x \in \mathbb{R}$. What are the examples of a cone other than the trivial one $\mathbb{R}^2$?

Answer 1

The cones are exactly the origin (more exactly, the set containing exactly the origin), the rays starting from the origin but not including it (in slight abuse of language, I'll call that an open ray), and arbitrary unions of those sets.

Proof:

• The origin is a cone:

  Obviously $\lambda(0,0)=(0,0)$.

• An open ray from the origin is a cone:

  The open ray from the origin passing through point $p=(x,y)\ne (0,0)$ is given by $C_p=\{\mu p:\mu \in \mathbb R^{>0}\}$. Thus if $\lambda\in\mathbb R^{>0}$ and $q\in C_p$, then there exists a $\mu>0$ such that $q=\mu p$, and then $\lambda q = \lambda\mu p\in C$ because $\lambda\mu>0$.

• The union of cones is a cone.

  Be $C_i$, $i\in I$, a collection of cones ($I$ is an arbitrary index set), and be $C=\bigcup_{i\in I}C_i$. Assume $p\in C$. Then there exists an $i\in I$ such that $p\in C_i$. But then for any $\lambda>0$, $\lambda p\in C_i$. But that implies $\lambda p\in C$.

• Any cone is such an union of open rays and possibly the origin.

  Be $C$ a cone. Be $p\in C$. Then either $p$ is the origin, or if not, then by definition of the cone, for all $\lambda>0$ we have $\lambda p\in C$. But the set $\{\lambda p: \lambda>0\}$ is exactly the open ray going through $p$, thus that open ray is a subset of $C$. Since through each $p\in C$ other than possibly the origin there's such a ray, it means $C$ is either the union of such rays (if the origin is not in $C$) or the union of those rays and the origin (if the origin is in $C$). $\square$

Note that arbitrary unions also include the empty union; indeed, the empty set vacuously fulfils the cone condition.

To add a few concrete examples of cones:

• The $x$ axis.

• The $x$ axis without the origin.

• The coordinate cross (union of $x$ axis and $y$ axis).

• Any of the four quadrants.

• The upper half-plane.

• The union of all straight lines through the origin whose slope is rational.

Answer 2

Cone of an arbitrary topological space $X$ can also be defined as the quotient space of $X \times [0,1]$ by identifying every point of $X \times \{1\}$ to a single point in $X \times \{1\}$ i.e $(x,1) \sim (y,1) \; \forall \; x,y \in X \times \{1\}$.

Observe that $X \times [0,1]$ is a cylinder. What we did is pinch all the points of one end of this cylinder say $X \times \{1\}$ to a single point in $X \times \{1\}.$

Now take any $X \subset \Bbb R$. You will get a cone in $\Bbb R^2$ as described above.