Descriptive and combinatorial set theory Introduction Singular - PowerPoint PPT Presentation

Descriptive and combinatorial set theory at singular cardinals and their successors Mirna Dˇ zamonja Descriptive and combinatorial set theory Introduction Singular cardinals, at singular cardinals and their successors descriptively Singular cardinals, combinatorially Singular cardinals, topologically Mirna Dˇ zamonja Successor of a singular cardinal School of Mathematics, University of East Anglia, associ´ ee IHPST, Universit´ e Panth´ eon-Sorbonne, Paris 1 Torino, September 2017

Descriptive and Generalising the Baire space combinatorial set theory at singular The Baire space ω ω is identified with the product ω ω and cardinals and their successors is given the usual product topology. Mirna Dˇ zamonja Introduction Singular cardinals, descriptively Singular cardinals, combinatorially Singular cardinals, topologically Successor of a singular cardinal

Descriptive and Generalising the Baire space combinatorial set theory at singular The Baire space ω ω is identified with the product ω ω and cardinals and their successors is given the usual product topology. A natural Mirna Dˇ zamonja generalisation of the space is a topology on κ κ for some Introduction κ > ℵ 0 and some natural generalisation of the product Singular cardinals, topology. descriptively Singular cardinals, combinatorially Singular cardinals, topologically Successor of a singular cardinal

Descriptive and Generalising the Baire space combinatorial set theory at singular The Baire space ω ω is identified with the product ω ω and cardinals and their successors is given the usual product topology. A natural Mirna Dˇ zamonja generalisation of the space is a topology on κ κ for some Introduction κ > ℵ 0 and some natural generalisation of the product Singular cardinals, topology. descriptively Singular cardinals, A natural generalisation of the product topology is to fix combinatorially some cardinal λ ≤ κ and to take basic open sets of the Singular cardinals, topologically form N ( f ) = { g : g ↾ dom ( f ) = f } for f a partial function Successor of a from κ to κ with | dom ( f ) | < λ . singular cardinal

Descriptive and Generalising the Baire space combinatorial set theory at singular The Baire space ω ω is identified with the product ω ω and cardinals and their successors is given the usual product topology. A natural Mirna Dˇ zamonja generalisation of the space is a topology on κ κ for some Introduction κ > ℵ 0 and some natural generalisation of the product Singular cardinals, topology. descriptively Singular cardinals, A natural generalisation of the product topology is to fix combinatorially some cardinal λ ≤ κ and to take basic open sets of the Singular cardinals, topologically form N ( f ) = { g : g ↾ dom ( f ) = f } for f a partial function Successor of a from κ to κ with | dom ( f ) | < λ . The most studied case is singular cardinal when λ = κ and it gives what people call the generalised Baire space .

Descriptive and Generalising the Baire space combinatorial set theory at singular The Baire space ω ω is identified with the product ω ω and cardinals and their successors is given the usual product topology. A natural Mirna Dˇ zamonja generalisation of the space is a topology on κ κ for some Introduction κ > ℵ 0 and some natural generalisation of the product Singular cardinals, topology. descriptively Singular cardinals, A natural generalisation of the product topology is to fix combinatorially some cardinal λ ≤ κ and to take basic open sets of the Singular cardinals, topologically form N ( f ) = { g : g ↾ dom ( f ) = f } for f a partial function Successor of a from κ to κ with | dom ( f ) | < λ . The most studied case is singular cardinal when λ = κ and it gives what people call the generalised Baire space . Topologists have studied combinatorial properties of generalised products of spaces since 1920s, usually with discouraging results, such as that the compactness is not preserved.

Descriptive and Generalising the Baire space combinatorial set theory at singular The Baire space ω ω is identified with the product ω ω and cardinals and their successors is given the usual product topology. A natural Mirna Dˇ zamonja generalisation of the space is a topology on κ κ for some Introduction κ > ℵ 0 and some natural generalisation of the product Singular cardinals, topology. descriptively Singular cardinals, A natural generalisation of the product topology is to fix combinatorially some cardinal λ ≤ κ and to take basic open sets of the Singular cardinals, topologically form N ( f ) = { g : g ↾ dom ( f ) = f } for f a partial function Successor of a from κ to κ with | dom ( f ) | < λ . The most studied case is singular cardinal when λ = κ and it gives what people call the generalised Baire space . Topologists have studied combinatorial properties of generalised products of spaces since 1920s, usually with discouraging results, such as that the compactness is not preserved. For example even the space R ω with the box topology is not connected or first countable, hence not metrisable.

Descriptive and Generalising the Baire space combinatorial set theory at singular The Baire space ω ω is identified with the product ω ω and cardinals and their successors is given the usual product topology. A natural Mirna Dˇ zamonja generalisation of the space is a topology on κ κ for some Introduction κ > ℵ 0 and some natural generalisation of the product Singular cardinals, topology. descriptively Singular cardinals, A natural generalisation of the product topology is to fix combinatorially some cardinal λ ≤ κ and to take basic open sets of the Singular cardinals, topologically form N ( f ) = { g : g ↾ dom ( f ) = f } for f a partial function Successor of a from κ to κ with | dom ( f ) | < λ . The most studied case is singular cardinal when λ = κ and it gives what people call the generalised Baire space . Topologists have studied combinatorial properties of generalised products of spaces since 1920s, usually with discouraging results, such as that the compactness is not preserved. For example even the space R ω with the box topology is not connected or first countable, hence not metrisable.

Descriptive and The generalised Baire space descriptively combinatorial set theory at singular Descriptive set theory of generalised spaces took longer cardinals and their successors to develop. Mirna Dˇ zamonja Introduction Singular cardinals, descriptively Singular cardinals, combinatorially Singular cardinals, topologically Successor of a singular cardinal

Descriptive and The generalised Baire space descriptively combinatorial set theory at singular Descriptive set theory of generalised spaces took longer cardinals and their successors to develop. The first paper on this subject was V¨ a¨ an¨ anen Mirna Dˇ zamonja (FM, 1991). Introduction Singular cardinals, descriptively Singular cardinals, combinatorially Singular cardinals, topologically Successor of a singular cardinal

Descriptive and The generalised Baire space descriptively combinatorial set theory at singular Descriptive set theory of generalised spaces took longer cardinals and their successors to develop. The first paper on this subject was V¨ a¨ an¨ anen Mirna Dˇ zamonja (FM, 1991). It considered the analogue of the Cantor-Bendixon theorem in ω 1 ω 1 and showed that its Introduction Singular cardinals, direct analogue (replacing ω by ω 1 ) is consistently true descriptively modulo a measurable cardinal. Singular cardinals, combinatorially Singular cardinals, topologically Successor of a singular cardinal

Descriptive and The generalised Baire space descriptively combinatorial set theory at singular Descriptive set theory of generalised spaces took longer cardinals and their successors to develop. The first paper on this subject was V¨ a¨ an¨ anen Mirna Dˇ zamonja (FM, 1991). It considered the analogue of the Cantor-Bendixon theorem in ω 1 ω 1 and showed that its Introduction Singular cardinals, direct analogue (replacing ω by ω 1 ) is consistently true descriptively modulo a measurable cardinal. It also introduced Singular cardinals, combinatorially connections with games. Singular cardinals, topologically Successor of a singular cardinal

Descriptive and The generalised Baire space descriptively combinatorial set theory at singular Descriptive set theory of generalised spaces took longer cardinals and their successors to develop. The first paper on this subject was V¨ a¨ an¨ anen Mirna Dˇ zamonja (FM, 1991). It considered the analogue of the Cantor-Bendixon theorem in ω 1 ω 1 and showed that its Introduction Singular cardinals, direct analogue (replacing ω by ω 1 ) is consistently true descriptively modulo a measurable cardinal. It also introduced Singular cardinals, combinatorially connections with games. Singular cardinals, topologically Today, the descriptive set theory of generalised Baire Successor of a spaces is well developed and involves many authors, singular cardinal including S. Friedman, Hyttinnen, Khomskii, Kulikov, Laguzzi, Motto Ros and many others.