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.

As $\mathbb{R}$ is homeomorphic to $(0,1)$, consider the cone on $(0,1)$. Now consider the open triangle with vertices $\{(0,0),(1,0),(1/2,1/2)\}$ union $(0,1)\times \{0\}$ in $\mathbb{R}\times [0,1]$. Its image in the cone is open which contains $(0,0)$ and whose closure is compact.
