# Is the quotient space of a contractible space contractible?

Let $X$ and $Y$ be topological spaces, where $X$ is contractible. Is the quotient space $X/Y$ contractible?

• Maybe Y is supposed to be contractible? Then the answer is positive if you work with CW complexes. I suspect that it is still negative in general. – Moishe Kohan Jun 2 '14 at 3:10

No. Consider $X=[0,1]$ and $Y=\{0,1\}$.