I have difficulty to solve this exercise of topology: Consider $X = \Bbb R$ − {$1$}, $Y = {y^2 = x^2 + x^3} ⊂ \Bbb R^2$ and $f : X → Y$, $f(t)=(t^2−1,t^3−t)$.Show that $f$ is continuous and bijective,that any $x∈X$ has a neighbourhood $U$ such that $f : U → f (U )$ is a homeomorphism.
I have difficulty to show the last part. I've that $f$ isn't closed therefore $f$ isn't an homeomorphism. Another question: when I show that function is continuous because $\Bbb R$ − {$1$} is disconnected also the nodal curve is disconnected but it seems connected to me. How can I show that the nodal curve is disconnected?
In fact it is connected. The point $(0, 0)$ would be the image of $-1$ and $1$, but removing $1$ still leaves $-1$. But the topology of the nodal curve doesn't change; it still topologically is a circle wedge a line. The circle is for $t$ between $-1$ and $1$, the wedge point is $(0, 0)$, and the line is from $(-\infty, -1]\cup(1, \infty)$. So this cannot be a homeomorphism. It's a continuous map as you said, but not bicontinuous.