Generic objects in topology Wiesaw Kubi s Institute of - PowerPoint PPT Presentation

Generic objects in topology Wiesław Kubi´ s Institute of Mathematics, Czech Academy of Sciences —— College of Science, Cardinal Stefan Wyszy´ nski University in Warsaw http://www.math.cas.cz/kubis/ 12th TOPOSYM Prague, 25-29.07.2016 W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 1 / 24

W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 2 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. BM ( K ) Fix a subcategory K of CM + , whose arrows are surjections. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. BM ( K ) Fix a subcategory K of CM + , whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. BM ( K ) Fix a subcategory K of CM + , whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K . Eve starts by choosing a K -object K 0 . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. BM ( K ) Fix a subcategory K of CM + , whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K . Eve starts by choosing a K -object K 0 . Odd responds by choosing a K -arrow k 1 0 : K 1 → K 0 . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. BM ( K ) Fix a subcategory K of CM + , whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K . Eve starts by choosing a K -object K 0 . Odd responds by choosing a K -arrow k 1 0 : K 1 → K 0 . Eve responds by choosing a K -arrow k 2 1 : K 2 → K 1 . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. BM ( K ) Fix a subcategory K of CM + , whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K . Eve starts by choosing a K -object K 0 . Odd responds by choosing a K -arrow k 1 0 : K 1 → K 0 . Eve responds by choosing a K -arrow k 2 1 : K 2 → K 1 . And so on... W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Denote by CM + the category of all nonempty compact metrizable spaces with continuous mappings. Consider the following infinite game. BM ( K ) Fix a subcategory K of CM + , whose arrows are surjections. Two players Eve and Odd alternately build an inverse sequence in K . Eve starts by choosing a K -object K 0 . Odd responds by choosing a K -arrow k 1 0 : K 1 → K 0 . Eve responds by choosing a K -arrow k 2 1 : K 2 → K 1 . And so on... The result is an inverse sequence k = � K i , k j � i , ω � . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 3 / 24

Who wins? W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins? BM ( K , U ) Fix a compact space U . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins? BM ( K , U ) Fix a compact space U . We say that Odd wins if the limit of the inverse sequence k 1 k 2 k 3 0 1 2 · · · K 0 K 1 K 2 is homeomorphic to U . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins? BM ( K , U ) Fix a compact space U . We say that Odd wins if the limit of the inverse sequence k 1 k 2 k 3 0 1 2 · · · K 0 K 1 K 2 is homeomorphic to U . Definition We say that U is K -generic if Odd has a winning strategy in BM ( K , U ) . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Who wins? BM ( K , U ) Fix a compact space U . We say that Odd wins if the limit of the inverse sequence k 1 k 2 k 3 0 1 2 · · · K 0 K 1 K 2 is homeomorphic to U . Definition We say that U is K -generic if Odd has a winning strategy in BM ( K , U ) . The game above will be called the Banach-Mazur game with parameters K and U . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 4 / 24

Uniqueness Proposition A K -generic compact space (if exists) is unique, up to homeomorphisms. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness Proposition A K -generic compact space (if exists) is unique, up to homeomorphisms. Proof. Suppose U 0 , U 1 are K -generic, witnessed by strategies Σ 0 , Σ 1 , 1 respectively. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness Proposition A K -generic compact space (if exists) is unique, up to homeomorphisms. Proof. Suppose U 0 , U 1 are K -generic, witnessed by strategies Σ 0 , Σ 1 , 1 respectively. Assume Odd uses Σ 0 . 2 W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness Proposition A K -generic compact space (if exists) is unique, up to homeomorphisms. Proof. Suppose U 0 , U 1 are K -generic, witnessed by strategies Σ 0 , Σ 1 , 1 respectively. Assume Odd uses Σ 0 . 2 Assume Eve uses Σ 1 . 3 W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness Proposition A K -generic compact space (if exists) is unique, up to homeomorphisms. Proof. Suppose U 0 , U 1 are K -generic, witnessed by strategies Σ 0 , Σ 1 , 1 respectively. Assume Odd uses Σ 0 . 2 Assume Eve uses Σ 1 . 3 They both win! 4 W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Uniqueness Proposition A K -generic compact space (if exists) is unique, up to homeomorphisms. Proof. Suppose U 0 , U 1 are K -generic, witnessed by strategies Σ 0 , Σ 1 , 1 respectively. Assume Odd uses Σ 0 . 2 Assume Eve uses Σ 1 . 3 They both win! 4 Thus U 0 ≈ U 1 . 5 W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 5 / 24

Universality Proposition Let U be a K -generic space. Then for every K ∈ K there exists a continuous surjection q : U → K. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 6 / 24

Universality Proposition Let U be a K -generic space. Then for every K ∈ K there exists a continuous surjection q : U → K. If all K -arrows are retractions, then q is a retraction. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 6 / 24

Basic examples Example The Cantor set 2 ω is CM + -generic. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 7 / 24

Basic examples Example The Cantor set 2 ω is CM + -generic. Proof. Odd has a simple winning tactic: At each step he chooses an arbitrary continuous surjection from 2 ω . W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 7 / 24

Basic examples Example The Cantor set 2 ω is CM + -generic. Proof. Odd has a simple winning tactic: At each step he chooses an arbitrary continuous surjection from 2 ω . Example Let Fin + be the category of nonempty finite sets with surjections. Then 2 ω is Fin + -generic. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 7 / 24

Continua Proposition There is no generic continuum. More precisely, there is no C -generic space, where C is the category of continua with continuous surjections. a continuum = a nonempty compact connected metrizable space W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 8 / 24

Continua Proposition There is no generic continuum. More precisely, there is no C -generic space, where C is the category of continua with continuous surjections. a continuum = a nonempty compact connected metrizable space Proof. Use Waraszkiewicz spirals. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 8 / 24

The pseudo-arc Notation: Denote by I the category whose unique object is the unit interval I = [ 0 , 1 ] and arrows are continuous surjections. W.Kubi´ s (http://www.math.cas.cz/kubis/) Generic objects in topology 26 July 2016 9 / 24