Higher-dimensional forcing -------------------- Bernhard Irrgang

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3 D forcing

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Three-dimensional applications

Let X be a topological space. Its spread is defined by

Theorem (Hajnal,Juhasz - 1967)
If X is a Hausdorff space, then card(X) ≤ 2^2spread(X) .

In his book ”Cardinal functions in topology”(1971), Juhasz explicitly asks if the second exponentiation is really necessary. This was answered by Fedorcuk (1975).

Theorem
In L, there exists a 0-dimensional Hausdorff (and hence regular) space with spread ω of size ω₂ = 2^2spread(X) .

This is a consequence of ♢ (and GCH).

There was no such example for the case spread(X) = ω₁. Three-dimensional forcing yields the following:

Theorem
If there is a simplified (ω₁,2)-morass, then there exists a ccc forcing of size ω₁ which adds a 0-dimensional Hausdorff space X of size ω₃ with spread ω₁.

Hence there exists such a forcing in L. By the usual argument for Cohen forcing, it preserves GCH. So the existence of a 0-dimensional Hausdorff space with spread ω₁ and size 2^2spread(X) is consistent.

Motivation

Two-dimensional applications

Open Problems

Morasses / Example