This relates to Munkres Topology, 2nd edition, exercise 1, chapter 7.44 Given n,

Question

This relates to Munkres Topology, 2nd edition, exercise 1, chapter 7.44

Given n, show that there is a continuous surjective map g: I --> I^n. [Hint: consider
f x f: I x I --> I^2 x 1^2] *I = [0, 1] of R.

I'm mainly interested in finding a continuous surjective map from I to I^3: a cube. Actually, I'd like to find a continuous surjective map from I to S^2 (sphere), but any regular polyhedron, like the cube, will do.

Any suggestions? I don't need to have a detailed, explicit formulation of such a function...but I do need to be able to make a strong argument that one can "fill" a cube with a curve. Do I start with the faces and work to the interior, i.e. extending the curve that fills a square, or do I need to take some other approach?
Although there are solutions to many problems in Munkres Topology, the solution to problem 1 was not among them.

Explanation / Answer

In situations like this, it helps if you remove the mental block that somehow a cube is "bigger" than an interval. See, the one is 3 dimensional and the other is 1 dimensional, but topologically speaking what is dimension? Sometimes it's just a mental block. So, if you can fill a square, then filling a cube is virtually the same thing. In fact, you could fill the cube with a square using quite an analogous method to that of filling the square with the interval. Then compose the two maps to get a map filling the cube with the interval. If you can increase dimension by one, all you have to do is repeat the process n times and then you can increase the dimension by n.

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