Is it possible for the following to be proof for Jordan curve theorem:
Given the distance function on $\mathbb{R}^2$ ($d((x_1,y_1),(x_2,y_2))=\sqrt{ |x_2-x_1|^2 + |y_2-y_1|^2}$), and $\varepsilon > 0$, let $X$, the inside of a topological circle with center $y$, be defined as $(x|d(x,y)<\varepsilon)$. You may define the outside of this circle, named the set $Y$ as $(x|d(x,y) > \varepsilon)$. The set Z, the outline of this circle is defined by $(x|d(x,y) = \varepsilon)$. This splits the circle into two parts, interior and exterior, or $X$ and $Y$, respectively, with a boundary of $Z$.
If you could prove homeomorphism between these circles and all Jordan curves (proving all Jordan curves have an interior and exterior) would this be a valid proof?
The Jordan-Schoenflies theorem states that the inside and outside of a Jordan curve are homeomorphic to the inside and outside of a standard circle in $\mathbb{R}^2$. You can read more in this paper.