The Tangent Disc Topology is developable

Question

A well-known example of a Moore space is the Tangent Disc Topology. I want to show that the Tangent Disc Topology is a developable space, i.e. it has a development. But I could not find the proof of it in any book. Could you give me any hint?

Thanks.

Answer 1

I’ll use the notation in the Wikipedia article on the tangent disk space $X$. For each $n\in\Bbb Z^+$ let

$$\mathscr{B}_n=\left\{U_{1/n}(p,q):\langle p,q\rangle\in\Bbb R\times\left(\frac1n,\to\right)\right\}\cup\left\{V_{1/n}(p,0):p\in\Bbb R\right\}\;;$$

I’ll leave it to you to show that $\{\mathscr{B}_n:n\in\Bbb Z^+\}$ is a development for $X$ and that $X$ is $T_3$.