Proved that a set is closed if and only if it contains all of its accumulation points.
Explained why the proof of the surjectivity of our function
f:[0,1] ---> solid equilateral triangle
boils down to proving that Image(f) is closed. To prove this we introduced the notion of a Hausdorff topological space, and noted that Euclidean space Rn is Hausdorff. Gave an example of a space that is not Hausdorff.
Ended with a proof of the result: In a Hausdorff topological space X any compact subspace A is closed.
This Youtube video shows a range of space-filling curves (including our "triangular" one).