I'm trying to prove that $\overline{st_K(x)}$ is a cone on $lk_K(x)$, but can't seem to get anywhere!
I know how to construct a topological cone given a space $X$. However I don't know any way to test whether a space is a cone on another space. Could someone give me a hint for this, and possibly outline some methods I might employ in the future for similar problems?
Many thanks!
Draw a picture. Alternately, refer to the picture at wikipedia. The easiest way to prove something is a cone based on some space is to figure out exactly what cone it is (in this case, the cone on $lk_K(x)$ from $x$ itself). In order to prove this you'll need to go to the definitions of the objects involved.