Difference between revisions of "Cone"

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{{:Finite dimensional continuum with C(X) homeomorphic to Cone(X) must have dimension 1}}
 
{{:Finite dimensional continuum with C(X) homeomorphic to Cone(X) must have dimension 1}}
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=Examples=
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#The [[harmonic fan]]

Revision as of 08:51, 5 April 2015

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

Theorem

Let $X$ be the circle with a spiral, then $C(X)$ is homeomorphic to $\mathrm{Cone}(X)$, where $C(X)$ denotes the hyperspace of continua of $X$.

Proof

References

Theorem

If $Y$ is a finite-dimensional continuum such that $C(X)$ is homeomorphic to $\mathrm{Cone}$$(X)$, then $\mathrm{dim}(X)=1$.

Proof

References

Examples

  1. The harmonic fan