Let \(X\) be a Hausdorff space. Show that if there is a continuous surjective map \(f: I \to X\), then \(X\) is compact, connected, weakly locally connected, and metrizable. [Hint: Show \(f\) is a perfect map.]
The converse of the result in (a) is a famous theorem of point-set topology called the Hahn-Mazurkiewicz theorem (see [H-Y], p. 129). Assuming this theorem, show there is a continuous surjective map \(f: I \to I^\omega\).
A Hausdorff space that is the continuous image of the closed unit interval is often called a Peano space.
Solution
- Let \(f:I\to X\) be continuous and surjective, with \(X\) Hausdorff. Since \(I\) is compact, \(X=f(I)\) is compact. Since \(I\) is connected, \(X\) is connected.
The map \(f\) is closed: a closed subset of \(I\) is compact, and its image is compact, hence closed in the Hausdorff space \(X\). Its fibers are closed subsets of \(I\), hence compact. Thus \(f\) is a perfect quotient map.
The interval \(I\) is locally connected, and a quotient space of a locally connected space is locally connected. Hence \(X\) is locally connected, and therefore weakly locally connected.
Finally, \(I\) is second countable, and perfect maps preserve second countability by E-Y4MFU(d). Thus \(X\) is second countable. A compact Hausdorff space is regular, so the Urysohn metrization theorem implies that \(X\) is metrizable.
Therefore every Hausdorff continuous image of \(I\) is compact, connected, weakly locally connected, and metrizable.
- The Hilbert cube \[ I^\omega=\prod_{n\ge1}I \] is compact by the Tychonoff theorem, connected because a product of connected spaces is connected, and metrizable because it is a countable product of metrizable spaces. It is also locally connected: a basic product neighborhood restricts only finitely many coordinates, and in those coordinates we can choose connected open intervals; the resulting product is connected and gives a connected neighborhood basis.
Hence \(I^\omega\) satisfies the hypotheses of the Hahn–Mazurkiewicz theorem. Therefore there exists a continuous surjection \[ I\twoheadrightarrow I^\omega. \]