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Applications of infinitary combinatorics 8


Academic year: 2021

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Applications of infinitary combinatorics 8


Kunen’s L-space [1981]


• HS = hereditarily separable,

• HL = hereditarily Lindelof,

• X - a Hausdorff space,

• K - the space constructed on the lecture,

• λ - the standard measure on 2ω1.

Zad. 1 Show that if X is metrizable, then X is HL ⇐⇒ X is HS.

Zad. 2 Show that if there is a continuous surjection f : X → [0, 1]ω1, then there is a non-separable measure on X.

Zad. 3 Show that whenever A is a countable subalgebra of Borel(2ω1), and B is a Borel subsets of 2ω1 such that λ(B) > 0, then there is a closed F ⊆ B such that

inf{λ(A4F ) : A ∈ A} > 0.

Zad. 4 Let F be the family of subsets used to extend the Boolean algebras in the construction of the space K, i.e.

F = {Fξ: ξ < ω1} ∪ {Fαξ: ξ, α < ω1}.

Show that for every x ∈ K

|{F ∈ F : x ∈ F }| = ℵ0.

Conclude that K is not separable (hint: otherwise K would have countable base). Hence, K is an L-space.

Zad. 5 Let (xα)α<ω1 be a well-ordered sequence of (different) real numbers. Denote Z = {xα: α < ω1}.

• Let τs be the topology on Z defined by neighbourhoods of points in the following way: the neighbourhoods of xα are of the form (xα− ε, xα+ ε) ∩ {xξ: ξ ≤ α}.

• Let τl be the topology on Z defined by neighbourhoods of points in the following way: the neighbourhoods of xα are of the form (xα− ε, xα+ ε) ∩ {xξ: ξ ≥ α}.

Show that (Z, τs) is an S-space, and (Z, τl) is an L-space. Show that those spaces are Hausdorff but not regular.




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