Fundamental group of theAlexandroff compactification of a subspace of $\mathbb R^2$

Question

Consider $\mathbb R^2$ with the euclidean topology and let $X=\{(x,y)\in \mathbb R^2:x^3-y=0\}$ be a subspace of the real plane.
(a) Is $X$ connected, locally compact and closed?
(b) Calculate the fundamental group of the compactification $\hat X=X\cup \{\infty\}$ with base point in $(0,0)$.

For the request (a) I had no problems, but I'm not sure about how approaching the second point of the exercise.
I'm used to computing fundamental groups (using van Kampen's theorems or via homotopic equivalence) of topological spaces like the torus , the Klein bottle, particular surfaces in $\mathbb R^2$ or in $\mathbb R^3$ and so on. I always worked with Alexandroff compactification showing, for example, certain topological properties of generic $\hat X$ spaces.
In this problem we have to compute the fundamental group of the compactification of $\Gamma_{y=x^3}$ and I'm not able (maybe it's easy) to visualize it, so I have difficulties in identifying the classes of homotopy of the loop that generate the topological space.

Thank for you for your help.

Answer 1

This community wiki solution is intended to clear the question from the unanswered queue.

Your question has been answered in the comments. The space $X$ is the graph of the function $f : \mathbb R \to \mathbb R, f(x) = x^3$. Thus $X \approx \mathbb R$.

(a) Yes.

(b) $\hat X \approx S^1$, thus $\pi_1(\hat X) \approx \mathbb Z$.