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On pseudointersections and condensers Piotr BorodulinNadzieja joint - - PowerPoint PPT Presentation

On pseudointersections and condensers Piotr BorodulinNadzieja joint with Grzegorz Plebanek UltraMath, Pisa June 2008 Piotr BorodulinNadzieja joint with Grzegorz Plebanek On pseudointersections and condensers Preliminaries Basic


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On pseudo–intersections and condensers

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek

UltraMath, Pisa

June 2008

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 2

Preliminaries

Basic remarks we will consider ultrafilters on P(N) and on Boolean subalgebras of P(N); if A is a subalgebra of P(N), then every ultrafilter on A is generated by a filter on P(N).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 3

Preliminaries

Basic remarks we will consider ultrafilters on P(N) and on Boolean subalgebras of P(N); if A is a subalgebra of P(N), then every ultrafilter on A is generated by a filter on P(N).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 4

Preliminaries

Basic remarks we will consider ultrafilters on P(N) and on Boolean subalgebras of P(N); if A is a subalgebra of P(N), then every ultrafilter on A is generated by a filter on P(N).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 5

Pseudo–intersection

Definition We say that P ⊆ N is a pseudo–intersection of a filter F if P \ F is finite (P ⊆∗ F) for every F ∈ F. Definition The pseudo–intersection number p is a minimal cardinality of a base of a filter without a pseudo–intersection. ℵ0 < p ≤ c; p = c under MA; p = ℵ1 < c in Sacks model (and many others).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 6

Pseudo–intersection

Definition We say that P ⊆ N is a pseudo–intersection of a filter F if P \ F is finite (P ⊆∗ F) for every F ∈ F. Definition The pseudo–intersection number p is a minimal cardinality of a base of a filter without a pseudo–intersection. ℵ0 < p ≤ c; p = c under MA; p = ℵ1 < c in Sacks model (and many others).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 7

Pseudo–intersection

Definition We say that P ⊆ N is a pseudo–intersection of a filter F if P \ F is finite (P ⊆∗ F) for every F ∈ F. Definition The pseudo–intersection number p is a minimal cardinality of a base of a filter without a pseudo–intersection. ℵ0 < p ≤ c; p = c under MA; p = ℵ1 < c in Sacks model (and many others).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 8

Pseudo–intersection

Definition We say that P ⊆ N is a pseudo–intersection of a filter F if P \ F is finite (P ⊆∗ F) for every F ∈ F. Definition The pseudo–intersection number p is a minimal cardinality of a base of a filter without a pseudo–intersection. ℵ0 < p ≤ c; p = c under MA; p = ℵ1 < c in Sacks model (and many others).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 9

Pseudo–intersection

Definition We say that P ⊆ N is a pseudo–intersection of a filter F if P \ F is finite (P ⊆∗ F) for every F ∈ F. Definition The pseudo–intersection number p is a minimal cardinality of a base of a filter without a pseudo–intersection. ℵ0 < p ≤ c; p = c under MA; p = ℵ1 < c in Sacks model (and many others).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 10

Asymptotic density

Definition The asymptotic density of a set A ⊆ N is defined as d(A) = lim

n→∞

|A ∩ [1, . . . , n]| n , provided this limit exists. The family {A: d(A) = 1} forms a filter

  • n N.

Definition For an infinite B = {b1 < b2 < b3 < . . .} ⊆ N define the relative density of A in B by dB(A) = d({n: bn ∈ A}) if this limit exists.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 11

Asymptotic density

Definition The asymptotic density of a set A ⊆ N is defined as d(A) = lim

n→∞

|A ∩ [1, . . . , n]| n , provided this limit exists. The family {A: d(A) = 1} forms a filter

  • n N.

Definition For an infinite B = {b1 < b2 < b3 < . . .} ⊆ N define the relative density of A in B by dB(A) = d({n: bn ∈ A}) if this limit exists.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 12

Asymptotic density

Definition The asymptotic density of a set A ⊆ N is defined as d(A) = lim

n→∞

|A ∩ [1, . . . , n]| n , provided this limit exists. The family {A: d(A) = 1} forms a filter

  • n N.

Definition For an infinite B = {b1 < b2 < b3 < . . .} ⊆ N define the relative density of A in B by dB(A) = d({n: bn ∈ A}) if this limit exists.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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Condenser

Definition Say that B ⊆ N is a condenser of a filter F on N if dB(F) = 1 for every F ∈ F. Remarks Every pseudo–intersection is a condenser; A density filter is an example of a filter with a condenser but without a pseudo–intersection.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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Condenser

Definition Say that B ⊆ N is a condenser of a filter F on N if dB(F) = 1 for every F ∈ F. Remarks Every pseudo–intersection is a condenser; A density filter is an example of a filter with a condenser but without a pseudo–intersection.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 15

Condenser

Definition Say that B ⊆ N is a condenser of a filter F on N if dB(F) = 1 for every F ∈ F. Remarks Every pseudo–intersection is a condenser; A density filter is an example of a filter with a condenser but without a pseudo–intersection.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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Another approach to condensation

Definition We say that a filter F on N is condensed if there is a bijection f : N → N such that d(f [F]) = 1 for every F ∈ F. Remarks if F has a condenser, then it is condensed; if F is condensed, then it is feeble, i.e. there is a finite–to–one function f : N → N such that f [F] is co–finite for every F ∈ F.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 17

Another approach to condensation

Definition We say that a filter F on N is condensed if there is a bijection f : N → N such that d(f [F]) = 1 for every F ∈ F. Remarks if F has a condenser, then it is condensed; if F is condensed, then it is feeble, i.e. there is a finite–to–one function f : N → N such that f [F] is co–finite for every F ∈ F.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 18

Another approach to condensation

Definition We say that a filter F on N is condensed if there is a bijection f : N → N such that d(f [F]) = 1 for every F ∈ F. Remarks if F has a condenser, then it is condensed; if F is condensed, then it is feeble, i.e. there is a finite–to–one function f : N → N such that f [F] is co–finite for every F ∈ F.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 19

Special Boolean algebras

Remarks it is quite easy to construct a subalgebra A of P(N) such that each ultrafilter on A does not have pseudo–intersection . . . . . . even if this A has to be small (i.e. does not contain uncountable independent family). Loosely speaking The more ultrafilters does not have a pseudo–intersection (condenser), the more rich has to be our algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 20

Special Boolean algebras

Remarks it is quite easy to construct a subalgebra A of P(N) such that each ultrafilter on A does not have pseudo–intersection . . . . . . even if this A has to be small (i.e. does not contain uncountable independent family). Loosely speaking The more ultrafilters does not have a pseudo–intersection (condenser), the more rich has to be our algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 21

Special Boolean algebras

Remarks it is quite easy to construct a subalgebra A of P(N) such that each ultrafilter on A does not have pseudo–intersection . . . . . . even if this A has to be small (i.e. does not contain uncountable independent family). Loosely speaking The more ultrafilters does not have a pseudo–intersection (condenser), the more rich has to be our algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 22

Problem

Problem Can we construct a subalgebra A of P(N) such that no ultrafilter on A has a pseudo–intersection; every ultrafilter on A is condensed? Answer - partial and easy assume CH; suppose no ultrafilter on A has a pseudo–intersection; then, it has to be 2c ultrafilters on A; thus, there is no enough bijections to ensure that every ultrafilter is condensed; conclusion: under CH there is no such an algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 23

Problem

Problem Can we construct a subalgebra A of P(N) such that no ultrafilter on A has a pseudo–intersection; every ultrafilter on A is condensed? Answer - partial and easy assume CH; suppose no ultrafilter on A has a pseudo–intersection; then, it has to be 2c ultrafilters on A; thus, there is no enough bijections to ensure that every ultrafilter is condensed; conclusion: under CH there is no such an algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 24

Problem

Problem Can we construct a subalgebra A of P(N) such that no ultrafilter on A has a pseudo–intersection; every ultrafilter on A is condensed? Answer - partial and easy assume CH; suppose no ultrafilter on A has a pseudo–intersection; then, it has to be 2c ultrafilters on A; thus, there is no enough bijections to ensure that every ultrafilter is condensed; conclusion: under CH there is no such an algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 25

Problem

Problem Can we construct a subalgebra A of P(N) such that no ultrafilter on A has a pseudo–intersection; every ultrafilter on A is condensed? Answer - partial and easy assume CH; suppose no ultrafilter on A has a pseudo–intersection; then, it has to be 2c ultrafilters on A; thus, there is no enough bijections to ensure that every ultrafilter is condensed; conclusion: under CH there is no such an algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 26

Problem

Problem Can we construct a subalgebra A of P(N) such that no ultrafilter on A has a pseudo–intersection; every ultrafilter on A is condensed? Answer - partial and easy assume CH; suppose no ultrafilter on A has a pseudo–intersection; then, it has to be 2c ultrafilters on A; thus, there is no enough bijections to ensure that every ultrafilter is condensed; conclusion: under CH there is no such an algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 27

Problem

Problem Can we construct a subalgebra A of P(N) such that no ultrafilter on A has a pseudo–intersection; every ultrafilter on A is condensed? Answer - partial and easy assume CH; suppose no ultrafilter on A has a pseudo–intersection; then, it has to be 2c ultrafilters on A; thus, there is no enough bijections to ensure that every ultrafilter is condensed; conclusion: under CH there is no such an algebra.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 28

Coefficient

What about the general result? Can we construct such an algebra in other models of ZFC? Definition A condensation number k is a minimal cardinality of a base of filter

  • n N without a condenser.

Facts p ≤ k; k ≤ b (a consequence of P. Simon’s result); consistently k < b (a consequence of M. Hrusak’s result).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 29

Coefficient

What about the general result? Can we construct such an algebra in other models of ZFC? Definition A condensation number k is a minimal cardinality of a base of filter

  • n N without a condenser.

Facts p ≤ k; k ≤ b (a consequence of P. Simon’s result); consistently k < b (a consequence of M. Hrusak’s result).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 30

Coefficient

What about the general result? Can we construct such an algebra in other models of ZFC? Definition A condensation number k is a minimal cardinality of a base of filter

  • n N without a condenser.

Facts p ≤ k; k ≤ b (a consequence of P. Simon’s result); consistently k < b (a consequence of M. Hrusak’s result).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 31

Coefficient

What about the general result? Can we construct such an algebra in other models of ZFC? Definition A condensation number k is a minimal cardinality of a base of filter

  • n N without a condenser.

Facts p ≤ k; k ≤ b (a consequence of P. Simon’s result); consistently k < b (a consequence of M. Hrusak’s result).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 32

Coefficient

What about the general result? Can we construct such an algebra in other models of ZFC? Definition A condensation number k is a minimal cardinality of a base of filter

  • n N without a condenser.

Facts p ≤ k; k ≤ b (a consequence of P. Simon’s result); consistently k < b (a consequence of M. Hrusak’s result).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 33

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 34

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 35

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 36

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

slide-37
SLIDE 37

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

slide-38
SLIDE 38

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

slide-39
SLIDE 39

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

slide-40
SLIDE 40

Construction of demanded algebra

Result (?) If k > h, then there is a Boolean algebra A such that no ultrafilter

  • n A has a pseudo–intersection, but each ultrafilter is condensed.

Sketch of proof consider a base matrix tree T (Balcar, Simon, Pelant); let B be a Boolean algebra generated by T ; there are two types of ultrafilters on B: branches and knots; B can be refined a little bit to an algebra A to ensure that knots are condensed; since k > h, every branch is condensed.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 41

Feebles

Theorem In the same manner it can be proved that if b > h (eg. in Hechler model), then there is a Boolean algebra A such that no ultrafilter on A has a pseudo–intersection; every ultrafilter on A is feeble.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 42

Dualities

Boolean algebra A compact space K = ult(A) Banach space X = C(K) dual Banach space M = C ∗(K) = M(K) . . .

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 43

Dualities

Boolean algebra A compact space K = ult(A) Banach space X = C(K) dual Banach space M = C ∗(K) = M(K) . . .

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 44

Dualities

Boolean algebra A compact space K = ult(A) Banach space X = C(K) dual Banach space M = C ∗(K) = M(K) . . .

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 45

Dualities

Boolean algebra A compact space K = ult(A) Banach space X = C(K) dual Banach space M = C ∗(K) = M(K) . . .

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 46

Dualities

Boolean algebra A compact space K = ult(A) Banach space X = C(K) dual Banach space M = C ∗(K) = M(K) . . .

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 47

Easy application

Fact Let p be an ultrafilter on a Boolean algebra A ⊆ P(N). The following conditions are equivalent: p has a pseudo–intersection {n1, n2, . . .}; lim nk = p in ult(A).

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 48

Gelfand–Phillips Property and Mazur property

Definition A Banach space X has a Mazur property if every weak∗–sequentially continuous x∗∗ ∈ X ∗∗ is continuous. A bounded subset A of a Banach space X is said to be limited if lim

n→∞ sup x∈A

|x∗

n(x)| = 0

for every weak∗–null sequence x∗

n ∈ X ∗.

Definition Banach space X has a Gelfand–Phillips property if every relatively norm compact space is limited.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 49

Gelfand–Phillips Property and Mazur property

Definition A Banach space X has a Mazur property if every weak∗–sequentially continuous x∗∗ ∈ X ∗∗ is continuous. A bounded subset A of a Banach space X is said to be limited if lim

n→∞ sup x∈A

|x∗

n(x)| = 0

for every weak∗–null sequence x∗

n ∈ X ∗.

Definition Banach space X has a Gelfand–Phillips property if every relatively norm compact space is limited.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 50

Gelfand–Phillips Property and Mazur property

Definition A Banach space X has a Mazur property if every weak∗–sequentially continuous x∗∗ ∈ X ∗∗ is continuous. A bounded subset A of a Banach space X is said to be limited if lim

n→∞ sup x∈A

|x∗

n(x)| = 0

for every weak∗–null sequence x∗

n ∈ X ∗.

Definition Banach space X has a Gelfand–Phillips property if every relatively norm compact space is limited.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 51

Mazur Property vs. Gelfand-Phillips Property

Problem It is known that there are Banach spaces with a Gelfand–Phillips property but without a Mazur property. Does Mazur property imply Gelfand–Phillips property? Fact If A is a Boolean algebra such that no ultrafilter on A has a pseudo–intersection but each ultrafilter on A has a condenser, then C(ult(A)) is an example of a Mazur space which does not possess the Gelfand–Phillips property.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 52

Mazur Property vs. Gelfand-Phillips Property

Problem It is known that there are Banach spaces with a Gelfand–Phillips property but without a Mazur property. Does Mazur property imply Gelfand–Phillips property? Fact If A is a Boolean algebra such that no ultrafilter on A has a pseudo–intersection but each ultrafilter on A has a condenser, then C(ult(A)) is an example of a Mazur space which does not possess the Gelfand–Phillips property.

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers

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SLIDE 53

The end

Slides and a preprint concerning the subject will be available on http://www.math.uni.wroc.pl/~ pborod

Piotr Borodulin–Nadzieja joint with Grzegorz Plebanek On pseudo–intersections and condensers