Is an infinite line the same thing as an infinite circle? Imagine that you are sitting next to a line that extends infinitely in both directions.  Is it possible to distinguish it from an infinite circle?
From my poor understanding of topology, I would guess that it makes a difference if it's a line or a circle: The second is closed, the first isn't.
What they both do to a plane is, that they split it into two parts, left and right or in and out. But is this enough to say that they are obviously the same thing?
EDIT
$\lim_{R\to \infty}$
$\hskip1in$

One sense in which a line segment and circle convergent to the same line for $R\to\infty$ is the following:
We will consider a vertical line segment going through the origin.
Similarly, let us take the circle going through $(0,0)$ such that the center lies on the $x$-axis. (I.e, the center will be $(R,0)$.) More precisely: 


*

*The line segment for some given $R$ is $L_R=\{0\}\times[-R,R]$.

*The circle for some given $R$ is $C_R=\{(x,y)\in\mathbb R^2; (x-R)^2+y^2=R^2\}$.


Then for $R\to\infty$ both these things obviously converge to the $y$-axis. (Maybe it helps if you draw a picture.)

If you want to replace word obviously in the last sentence by something more rigorous, you can take Kuratowski convergence.
This means that you can ask: What is the set  $$L=\{p\in\mathbb R^2; \lim_{R\to\infty} d(p,L_R)=0\}$$
where $d(p,A)$ denotes the distance of the point $p$ and the set $A$.
Similarly, you can ask about $$C=\{p\in\mathbb R^2; \lim_{R\to\infty} d(p,C_R)=0\}.$$
You will find out that both $L$ and $C$ are equal to $y$ axis.

Perhaps the following picture might illustrate how the circles get closer and closer to the vertical line with increasing radius.

