Cone

From hyperspacewiki
Jump to: navigation, search

Let $(X,\tau)$ be a topological space. We define the cone over $X$ taking $X \times [0,1]$ and shrinking $X \times \{1\}$ to a point. More precisely we define $\mathrm{Cone}(X)$ to be the quotient space $X \times [0,1] / \sim$, where $\sim$ is an equivalence class defined by $(x_1,t_1) \sim (x_2,t_2)$ if and only if $t_1=t_2=1$. The point $X \times \{1\} \in \mathrm{Cone}(X)$ is called the vertex of the cone while $X \times \{0\}$ is called the base of $\mathrm{Cone}(X)$.

Properties

Cone over circle with spiral homeomorphic to hyperspace of continua of circle with spiral
Finite dimensional continuum with C(X) homeomorphic to Cone(X) must have dimension 1

Examples

  1. The harmonic fan