How to find centered subsets in the forcing Hechler

Question

Let Hechler forcing $\mathbb{D}$. Define $\mathbb{D}=\omega^{<\omega }\times{}\omega^{\omega }$( not cardinal arithmetic) ordered as $(t,g)$ iff $s \subseteq t$ and $f \leq g$ (that is, $\forall_{n<\omega }(f(n)\leq g(n)) )$ and $\forall_{i \in |t|\setminus |s|}(f(i) \leq t(i))$.

How to find centered subsets $\left<{Q_{s}}\right>_{s \in{\omega^{<\omega}}}$ such that $\mathbb{D}=\bigcup_{s \in{\omega^{<\omega}}}Q_{s}$.?

a suggestion please for this problem.

Thanks.

Answer 1

I’m assuming that you accidentally omitted a bit of the definition of $\Bbb D$, and that the ordering is that $\langle s,f\rangle\preceq\langle t,g\rangle$ iff

• $s\subseteq t$,
• $f\le g$, and
• $f(k)\le t(k)$ for each $k\in|t|\setminus|s|$.

For each $s\in{^{<\omega}\omega}$ let $Q_s=\{\langle s,f\rangle:f\in{^\omega\omega}\}\subseteq\Bbb D$; I claim that each $Q_s$ is centred. Suppose that $F=\{\langle s,f_i\rangle:i\in n\}$ is a finite subset of $Q_s$; you need to show that there is a condition $\langle s,g\rangle\in Q_s$ such that $\langle s,g\rangle\preceq\langle s,f_i\rangle$ for each $i\in n$. The first and third bulleted conditions are trivially satisfied, and you should have no trouble finding a $g\in{^\omega\omega}$ that satisfies the second for each $f_i$.