Is every product (finite or infinite) of Hausdorff spaces Hausdorff?
Is \(\RR\) homeomorphic to \([0, \infty)\)?
Solution
1. Every product, finite or infinite, of Hausdorff spaces is Hausdorff.
If two points \(x=(x_i)\) and \(y=(y_i)\) differ, choose a coordinate \(j\) with \(x_j\ne y_j\). Separate \(x_j,y_j\) by disjoint open sets \(U,V\) in the Hausdorff factor \(X_j\). Their inverse images under the projection \(\prod_iX_i\to X_j\) are disjoint open neighborhoods of \(x,y\).
2. The spaces \(\mathbb R\) and \([0,\infty)\) are not homeomorphic.
Removing any point from \(\mathbb R\) disconnects it into two components. But removing the endpoint \(0\) from \([0,\infty)\) leaves \((0,\infty)\), which is connected. A homeomorphism would preserve this point-deletion property.