Suspension of Hausdorff space is Hausdorff

Statement
Suppose $$X$$ is a fact about::Hausdorff space. Then the fact about::suspension of $$X$$, commonly denoted $$SX$$, is also a proves property satisfaction of::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 $$p_1$$:
 * 2) Case that one of $$a$$ or $$b$$ is equal to $$p_0$$ or $$p_1$$: