Homogeneous spaces and Wadge theory Sandra M uller Universit at - PowerPoint PPT Presentation

Homogeneous spaces and Wadge theory Sandra M¨ uller Universit¨ at Wien January 2019 joint work with Rapha¨ el Carroy and Andrea Medini Arctic Set Theory Workshop 4 Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 1

How I got interested in general topology Our main tool: Wadge theory The beauty of Hausdorff operations Putting everything together Open questions and future goals Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 2

How I got interested in general topology Our main tool: Wadge theory The beauty of Hausdorff operations Putting everything together Open questions and future goals Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 3

Homogeneous spaces All our topological spaces will be separable and metrizable. A homeomorphism between two spaces X and Y is a bijective continuous function f such that the inverse f − 1 is continuous as well. Definition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h : X → X such that h ( x ) = y . Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 4

Homogeneous spaces All our topological spaces will be separable and metrizable. A homeomorphism between two spaces X and Y is a bijective continuous function f such that the inverse f − 1 is continuous as well. Definition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h : X → X such that h ( x ) = y . X X h x y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 4

Homogeneous spaces All our topological spaces will be separable and metrizable. A homeomorphism between two spaces X and Y is a bijective continuous function f such that the inverse f − 1 is continuous as well. Definition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h : X → X such that h ( x ) = y . X X h x y Examples of homogeneous spaces: all discrete spaces, Q , 2 ω , ω ω ≈ R \ Q , all topological groups. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 4

Homogeneous spaces Definition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h : X → X such that h ( x ) = y . Examples of homogeneous spaces: all discrete spaces, Q , 2 ω , ω ω ≈ R \ Q , all topological groups. We will focus on zero-dimensional homogeneous spaces, i.e. topological spaces which have a base of clopen sets. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 4

Homogeneous spaces Definition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h : X → X such that h ( x ) = y . Examples of homogeneous spaces: all discrete spaces, Q , 2 ω , ω ω ≈ R \ Q , all topological groups. We will focus on zero-dimensional homogeneous spaces, i.e. topological spaces which have a base of clopen sets. Fact X is a locally compact zero-dimensional homogeneous space iff X is discrete, X ≈ 2 ω , or X ≈ ω × 2 ω . We will therefore focus on non-locally compact (equivalently, nowhere compact) zero-dimensional homogeneous spaces. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 4

h-homogeneity Definition A space X is h-homogeneous if every non-empty clopen subset U of X (with the subspace topology) is homeomorphic to X . Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 5

h-homogeneity Definition A space X is h-homogeneous if every non-empty clopen subset U of X (with the subspace topology) is homeomorphic to X . Fact A zero-dimensional space X is h-homogeneous iff for all non-empty clopen proper subsets U, V of X there is a homeomorphism h : X → X such that h [ U ] = V . X X h U V Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 5

h-homogeneity Definition A space X is h-homogeneous if every non-empty clopen subset U of X (with the subspace topology) is homeomorphic to X . Fact A zero-dimensional space X is h-homogeneous iff for all non-empty clopen proper subsets U, V of X there is a homeomorphism h : X → X such that h [ U ] = V . X X Examples of h-homogeneous spaces: h U V Q , 2 ω , ω ω , any product of zero-dimensional h-homogeneous spaces (Medini, 2011) Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 5

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. x y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. x y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. x y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. x y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. h 0 x x y y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. h 0 h 1 x x x y y y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. Proof by picture. h 0 h 1 x x x y y y Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. h 0 h 1 x x x y y y n ∈ ω ( h n ∪ h − 1 Now � n ) can be extended to a homeomorphism h : X → X such that h ( x ) = y and h − 1 ( y ) = x . Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. But the converse does not hold in general. Theorem (van Mill, 1992) ( AC ) There exists a zero-dimensional homogeneous space that is not h-homogeneous. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. But the converse does not hold in general. Theorem (van Mill, 1992) ( AC ) There exists a zero-dimensional homogeneous space that is not h-homogeneous. Theorem (van Engelen, 1986) A Borel non-locally-compact subspace of 2 ω is homogeneous if and only if it is h-homogeneous. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

h-homogeneity versus homogeneity Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous. But the converse does not hold in general. Theorem (van Mill, 1992) ( AC ) There exists a zero-dimensional homogeneous space that is not h-homogeneous. Theorem (van Engelen, 1986) A Borel non-locally-compact subspace of 2 ω is homogeneous if and only if it is h-homogeneous. Question Can we say more under projective determinacy ( PD ) or AD ? Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 6

Yes! Theorem (Carroy – Medini – M) ( PD ) A projective non-locally-compact subspace of 2 ω is homogeneous if and only if it is h-homogeneous. ( AD + DC ) A non-locally-compact subspace of 2 ω is homogeneous if and only if it is h-homogeneous. Sandra M¨ uller (Universit¨ at Wien) Homogeneous spaces and Wadge theory January 2019 7