Why do equatons of two variables specify curves in $\mathbb{R}^2$? I suppose to more formally characterize the question more formally, why are all points of the set $\{ (x,y) \mid F(x,y) = 0 \}$ always boundary points (and I believe also never isolated points) in the standard topology of $\mathbb{R}^2$ -- for simplicity's sake, say in the case where $F(x,y) \in \mathbb{R}[x,y]$ (i.e. polynomials in $x,y$ with coefficients in $\mathbb{R}$)?
I imagine there are standard proofs of such a proposition (and certainly more general than this) in algebraic geometry, but I have not studied algebraic geometry, and was wondering if there were any elementary proofs of this, only relying on basic point-set topology, algebra, real analysis, etc... I'm also looking for a relatively intuitive explanation/proof, if possible. Maybe the most intuitive reason why this is true is because of the implicit function theorem, but I wonder if there are other, still intuitive, elementary proofs of this.

If "intuitive" is what you want, then think of what the graph of a function of two variables looks like.  Hills and valleys.  If you have $F(x,y) = \text{specified constant}$, then the value of the constant tells you to what depth you fill fill the world with water, and then the curve is the coastline. Why is the coastline a curve?  That is the question. 
