Suspension of Hausdorff space is Hausdorff

Statement

Suppose $X$ is a Hausdorff space (?). Then the Suspension (?) of $X$, commonly denoted $SX$, is also a Hausdorff space.

Proof

Given: A Hausdorff space $X$. The suspension $SX$ is defined as the quotient of $X\times [0,1]$ by the collapse of $X\times \{0\}$ and $X\times \{1\}$ to single points, which we call $p_{0}$ and $p_{1}$.

To prove: $SX$ is Hausdorff.

Proof: Suppose $a$ and $b$ are two distinct points in $SX$. We consider two cases:

1. Case both $a$ and $b$ are not equal to either $p_{0}$ or $p_{1}$: Fill this in later
2. Case that one of $a$ or $b$ is equal to $p_{0}$ or $p_{1}$: Fill this in later