We have now seen four different proofs of the Fundamental Theorem of Algebra. The first two were purely analysis, while the second pair involved a wide collection of algebraic ideas. However, we should realize that even in these algebraic proofs we did not totally leave analysis. Each of these proofs used the fact that odd-degree real polynomials have real roots. This fact is a consequence of the intermediate value theorem, which depends on continuity. Continuity is a topological property and we now proceed to our final pair of proofs, which involve topology.