Is the cone locally compact Let $X$ denote the cone on the real line $\mathbb{R}$. Decide whether $X$ is locally
compact. [The cone on a space $Y$ is the quotient of $Y \times I$ obtained by
identifying $Y \times \{0\}$ to a point.]
I am having a hard time showing that there exists a locally compact neighborhood around $Y \times \{0\}$.  Some help would be nice.

Hint: $X$ is homeomorphic to $C= \{(x,y,z) \mid z \geq 0, x^2+y^2=z^2 \}$ and $C_n= \{ (x,y,z) \mid 0 \leq z \leq 1/n,  x^2+y^2=z^2 \}$ is a compact neighborhood of $(0,0,0)$.
