Let \(X\) be the letter \(\mathsf{X}\) in \(\mathbb{R}^2\text{,}\) viewed as the union of the two line segments from \((-1,-1)\) to \((1,1)\) and from \((-1,1)\) to \((1,-1)\text{.}\)
Show that the Mรถbius band deformation retracts onto its central circle. (You may describe the Mรถbius band as the quotient \([0,1]\times[-1,1]\big/{\sim}\) where \((0,y)\sim(1,-y)\text{,}\) and the central circle as the image of \([0,1]\times\{0\}\text{.}\) Construct an explicit deformation retraction.)
Let \(f,g\colon X\to Y\) be homotopic maps and let \(r\colon Y\to Z\) be a continuous map. Prove that \(r\circ f\simeq r\circ g\text{.}\) (That is, post-composition preserves the homotopy relation.)
Let \(\sigma\colon\Delta^2\to X\) be a singular \(2\)-simplex. Write out the singular \(0\)-chain \(\partial_1(\partial_2(\sigma))\) explicitly as a sum of singular \(0\)-simplices (with signs), and verify directly that it equals zero.
Let \(S=\{a,b\}\) be the Sierpiลski space: the topology is \(\{\varnothing,\{a\},\{a,b\}\}\text{.}\) Compute \(H_0(S)\) directly from the definition. (Determine all continuous maps \(\Delta^1\to S\text{,}\) use them to find the group of \(0\)-boundaries \(B_0\text{,}\) and compute the quotient \(Z_0/B_0\text{.}\))
