show/hide this revision's text 4 "well known" to Tall in 1994

I am looking for a proof (or better, a reference) of the following fact:

The finite support iteration of $\sigma$-centered forcing notions is again $\sigma$-centered, assuming we iterate less than $(2^{\aleph_0})^+$ steps.

(EDIT: In the first version of the question I forgot to mention that it was Stefan Geschke who suggested that there should be a proof similar to "the product of continuum many separable spaces is still separable")

EDIT: In his 1994 paper "$\sigma$-centred forcing and reflection of (sub)metrizability" in TAMS (MR1179593 94g:54003), Frank Tall writes:

"It is well known (proved by the same method that proves the product of $\le 2^{\aleph_0}$ separable spaces is separable) that the finite support iteration of $\le 2^{\aleph_0}$ $\sigma$-centred orders is $\sigma$-centred."

show/hide this revision's text 3 Geschke

I am looking for a proof (or better, a reference) of the following fact:

The finite support iteration of $\sigma$-centered forcing notions is again $\sigma$-centered, assuming we iterate less than $(2^{\aleph_0})^+$ steps.

(Assuming EDIT: In the first version of the question I forgot to mention that it is true. Isn't was Stefan Geschke who suggested that there should be a proof using the fact that similar to "the product of continuum many separable spaces is still separable? Or at least using the same idea.)

(A counterexample would be welcome, too.separable")
show/hide this revision's text 2 or counterexample

I am looking for a proof (or better, a reference) of the following fact:

The finite support iteration of $\sigma$-centered forcing notions is again $\sigma$-centered, assuming we iterate less than $(2^{\aleph_0})^+$ steps.

(Assuming it is true. Isn't there a proof using the fact that the product of continuum many separable spaces is still separable? Or at least using the same idea.)

(A counterexample would be welcome, too.)

show/hide this revision's text 1