In the instance of a line $L$ intersecting with a parabola I need to find out if the intersection is a dense subset. Here's my work / question so far:
For another point $P$ on the parabola not in the intersection of $L$, is the entire complement of another variety which contains that point counted as a neighborhood or just the connected part around the point? That is whenever I look at a circle $C$ centered at $P$ which doesn't have the intersection of the line within the radius then is the complementary set of $C$ which has two parts- one which has the point in it and the other with the line intersecting the parabola counted as a neighborhood around $P$ which contains the intersection of the line? The text 'An Invitation to Algebraic Geometry' does say that the open sets are very large so can't that just mean that the set is dense along with that the two parts in and outside of the circular variety are connected, because they can not really be complementary sets of any polynomials!
From your question you seem to be slightly confused about what the Zariski topology looks like. We do not want to look at "open balls of radius $r$" like we would in the usual euclidean topology, instead we want to be looking at zero sets of polynomials.
Recall that a set $Y \subset X$ is dense if and only if $\bar{Y} = X$. Now, the line $L$ and the parabola $X$ are both closed subsets of $\mathbb{A}^2$ since they are the vanishing sets of some polynomials $y - mx - c$ and $y - x^2 - ax - b$ respectively. Thus the intersection $L \cap X$ is a proper closed subset of $X$ hence not dense.