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Given a topological space $X$ consider the quotient space $X \times [0,1]$ such that all points $(x,0)$ are identified to a single point $p$. This space $CX/X \times 0$ is called the cone over $X$.

Suppose $X$ is a compact subset of $\mathbb{R}^n$ for some $n$. View $\mathbb{R}^n$ as a subset of $\mathbb{R}^{n+1}$ in the usual way. Choose a point $x_o \in \mathbb{R}^{n+1}-\mathbb{R}^n$. Let $C$ be the subspace of $\mathbb{R}^{n+1}$ consisting of the union of all line segments from $x_0$ to points in $X$. Show that $C$ is homeomorphic to the cone over $X$ as defined above.

I am going to try and construct a map $f:CX \rightarrow C$ and use the universal quotient property as described in this problem to show it induces a homeomorphism $g:CX/X \times 0\rightarrow C$.

How to use universal quotient property to construct homeomorphism

Define $f:CX \rightarrow C$ by $f(x,t)=x_ot+(1-t)x$.

Now I am concerned with how to show $(x',t') \sim (x,t) \implies f(x,t) \sim f(x',t')$.

Suppose $(x,t) \sim (x',t')$. Then $x_ot+(1-t)x=x_ot'+(1-t')x'$ or $t=t'=0$ and $x=x'$.

Am I on the right track with this? If so how should I finish it up to prove it is a homeomorphism?

So by the universal quotient property $g:CX/X \times 0 \rightarrow C$ is continuous.

ernesto
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The right notation is $CX := X \times [0,1]{/}(X \times \{0\}$.

Define $f: X \times [0,1] \to C$ by $f(x,t)= (1-t)x_0 + tx \in \Bbb R^{n+1}$, which is continuous for simple reasons.

Then $f(x,t)= f(x',t') \iff (x,t)=(x',t') \lor (t=t'=0)$ and so $f$ exactly respects the equivalence relation that identifies $X \times \{0\}$ to a point.

So the induced map $\tilde{f}: CX \to C$ is a homeomorphism by compactness and Hausdorffness.

Henno Brandsma
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