Is the 1-dimension projective line a circle? Is the 1 dimensional projective line homeomorphic to the circle?  If so, the circle is homeomorphic to itself with antipodal points identified  (very unintuitive).  I am missing something?
 A: Yes, the real $1$-dimensional projective line $\mathbb{RP}^1$ is homeomorphic (in fact diffeomorphic) to the circle $S^1$. It can be thought of as the circle $S^1$ with antipodal points identified, which reflects the fact that the circle is its own double cover: the double cover map is given explicitly by
$$S^1 \ni z \mapsto z^2 \in S^1$$
thinking of $S^1$ as the unit complex numbers. In other words, quotienting by antipodes is the same as quotienting by $180^{\circ}$ rotation; note that this is very much false for the 2-sphere. 
A: Here is a somewhat lower-brow 'proof by picture' of how such a process of gluing might go. To save you from having to read my poor handwriting, the steps are:
1) Embed $S^1$ into the $(x,y)$-plane in $\mathbb{R}^3$.
2) Pick an antipodal pair and glue (together) to the $z$-axis. 
3) Pick another pair of non-glued antipodal points, and snip the two loops at this pair.
4) Glue the antipodal pairs of line segments (together) along the $z$-axis.  You'll have but one pair left unglued.
5) Glue final pair.

