/sci/ - Science & Math

The surface of a hypercube is homeomorphic to the 3-sphere via, say, [math]f[/math] from the sphere to the surface. Suppose you could have a continuous injection [math]g[/math] from the hypercube's surface to [math]\mathbb{R}^2[/math], and let [math]i \colon \mathbb{R}^2 \to \mathbb{R}^3[/math] be the inclusion [math]x \mapsto (x, 0)[/math]. Define [math]h=i \circ g \circ f[/math]. Then [math]h \colon S^3 \to \mathbb{R}^3[/math] is a continuous injection contradicting Borsuk-Ulam.