Show that \((X_1 \times \cdots \times X_{n-1}) \times X_n\) is homeomorphic with \(X_1 \times \cdots \times X_n\).
Solution
Write \(P=X_1\times\cdots\times X_{n-1}\) with projections \(p_i\colon P\to X_i\), let \(q_1\colon P\times X_n\to P\) and \(q_2\colon P\times X_n\to X_n\) be the projections, and let \(\pi_i\colon X_1\times\cdots\times X_n\to X_i\) be the projections. Define \[ h\colon P\times X_n\to X_1\times\cdots\times X_n,\qquad h((x_1,\ldots,x_{n-1}),x_n)=(x_1,\ldots,x_n). \]
1. \(h\) is a continuous bijection.
It has inverse \(k(x_1,\ldots,x_n)=((x_1,\ldots,x_{n-1}),x_n)\). Its coordinates are \(\pi_i\circ h=p_i\circ q_1\) for \(i<n\) and \(\pi_n\circ h=q_2\), composites of continuous projections; a map into a product is continuous if and only if its coordinates are.
2. \(k=h^{-1}\) is continuous.
Its coordinates are \(q_1\circ k=(\pi_1,\ldots,\pi_{n-1})\colon X_1\times\cdots\times X_n\to P\), continuous because its coordinates \(\pi_i\) are, and \(q_2\circ k=\pi_n\).
3. Q.E.D.
By steps 1 and 2, \(h\) is a homeomorphism.