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Start with the collection of half-open intervals of the form $[a,a+1) \times 0$ where $a \geq 0$ is an integer. This decomposes the positive $x$-axis into half-open intervals. Now, for every value of $0 < \theta < 2\pi$, decompose the ray whose angle with the positive $x$-axis is $\theta$ into half-open intervals with the open end of the interval placed at the endpoint nearest the origin.