fixed point theorem

**selenne431** · March 8th 2010, 09:30 PM

Prove that for every function $f: \omega_1 \rightarrow \omega_1$ , there is a $\beta \in \omega_1$ such that $f[\beta] \subseteq \beta$ .

Hint: A fixed point theorem may be useful. We may us the Countable Principle of Choice, but not the Axiom of Choice.

**emakarov** · March 9th 2010, 04:59 AM

Are you referring to Knaster–Tarski fixpoint theorem? Since it requires a monotonic function, maybe one can consider $g(\beta)=\bigcup_{\alpha\le\beta}f(\alpha)$ . Then $g$ is monotonic, so it has a fixpoint $\beta_0$ : $g(\beta_0)=\beta_0$ , which implies $f(\beta_0)\subseteq\beta_0$ . And the Countable Principle of Choice is used to show that $\omega_1$ is a complete lattice. I am not sure about this, but it may be a start...

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