Let $X = (-1,1)$ be considered with the Euclidean metric, and $Y = (0, \infty)$ be given the cofinite topology. Prove that $X$ and $Y$ are not homeomorphic.
My current thoughts are that a homeomorphism is a continuous bijection with a continuous inverse, and that it's relatively trivial to define a bijection between $(-1,1)$ and $(0,\infty)$, so I need to show that any function between $X$ and $Y$ is not continuous due to the topologies. This would be done by showing that if $f:X\rightarrow Y$ is a bijective function, and a set $A$ is open in $X$, then $f^{-1}(A)$ is closed in $Y$. Here is where I hit a wall and am unable to continue, any help would be appreciated!
Best Answer
HINT: One of the spaces is Hausdorff, and the other is not.