This is a quandary that came to mind while pondering the question What is the mathematical definition of a "figure six"? .
The following is my effort to formalize the idea of two curves "merging tangentially" and becoming "infinitely close without touching". In the following, one of those curves is the line $\mathscr{L}$.
Suppose I define a smooth plane curve $\mathscr{C}$ as an open set of points such that the limit of the angle between the tangent to $\mathscr{C}$ and some line $\mathscr{L}$ approaches $0$ as the distance $d[\mathbf{p},\mathscr{L}]:\mathbf{p}\in\mathscr{C}$ between a variable point on the curve and $\mathscr{L}$ approaches $0$.
Assert $\mathscr{C}\cap\mathscr{L}=\emptyset$.
In addition, let there be a line, $\mathscr{L}_{\perp}$ perpendicular to $\mathscr{L}$, and some neighborhood, $\mathscr{A}$ of the point of intersection of $\mathscr{L}$ and $\mathscr{L}_{\perp}$. Also, one end of $\mathscr{C}$ shall lie outside of $\mathscr{A}$.
Within $\mathscr{A}$, require for some arbitrarily small, fixed $\varepsilon>0$, that
$$0<d[\mathbf{p},\mathscr{L}_\perp]<d[\mathbf{p},\mathscr{L}]<d[\mathbf{p},\mathscr{L}_{\perp}](1+\varepsilon):\mathbf{p}\in\mathscr{A}\cap\mathscr{C}.$$
Does this consistently describe a smooth, open curve with a boundary point at the intersection of $\mathscr{L}$ and $\mathscr{L}_\perp$, but which does not include that boundary point?
Now, if $\mathscr{L}$ is replaced by a colinear segment which also intersects $\mathscr{L}_\perp$ and smoothly continues the erstwhile end of the curve $\mathscr{C}$ at some point outside of $\mathscr{A}$, is the end of $\mathscr{C}$ within $\mathscr{A}$ now closed? In other words, now that the point $\mathscr{L}\cap\mathscr{L}_\perp\in\mathscr{C}$.