Let $X = \mathbb{R}^2 - J$ where $J$ is a Jordan curve. We know that $X = \bigcup_{i \in \mathcal{I}}$ where the subsets $U_{i}$ are the path components of $X.$ These subsets are pairwise disjoint. (we do not assume that the topology on $X$ is the disjoint union topology)
A Jordan curve in $\mathbb{R}^2$ is the image $J$ of an injective continuous function $w : S^1 \rightarrow \mathbb{R}^2.$ If $J$ is a Jordan curve with the inclusion map $j : J \rightarrow \mathbb{R}^2,$ then there is the following commutative diagram > $$\require{AMScd} \begin{CD} S^1 @>{w}>> \mathbb{R}^2\\ @VVV @VVV \\ J @>{j}>> \mathbb{R}^2 \end{CD}$$
The first vertical arrow should have the map $\bar{\omega}$ on it. Also,I am not skillful in drawing commutative diagrams this is why I draw $\mathbb{R}^2$ 2 times because I do not know how to draw one curved arrow coming out of $J$ going directly to $\mathbb{R}^2$ my bad.
Now using the compactness of $S^1$ and the Hausdorffitude of $\mathbb{R}^2,$ we can see that $\bar{\omega} : S^1 \rightarrow J$ is a homeomorphism.
having already knew the answers of those questions with the help of @jgon:
$(a)$ Show that $X$ is open in $\mathbb{R}^2.$
$(b)$ Show that each $U_{i}$ is open in $\mathbb{R}^2.$
I was given a hint to assume (without proving it) that every open ball $B_{r}(\mathbf{x})$ in $\mathbb{R}^2$ is convex.
My questions are:
$(c)$ Show that $J\subseteq B_{R}(\mathbf{0})$ for some $R.$
$(d)$ Show that every path component of $X$ except one is contained in $B_{R}(\mathbf{0})$ (where $R$ is the same number as in part $(c)$).
for this question I was given the following hint: think in terms of the equivalence relation $\sim$ defined by $x \sim y$ iff there is a path from $x$ to $y.$
Still I am unable to answer $(c)$ & $(d)$ could anyone give me a hint for answering them please?