Reversible spaces STDC, Baylor University. Rodrigo Hernández-Gutiérrez Nipissing University Joint work with Alan Dow

Transcription

1 Reversible spaces 2016 STDC, Baylor University Rodrigo Hernández-Gutiérrez Nipissing University Joint work with Alan Dow March 12, 2016

2 Definition A topological space X is reversible if every continuous bijection f : X X is a homeomorphism.

3 Definition A topological space X is reversible if every continuous bijection f : X X is a homeomorphism. Some examples of reversible spaces are:

4 Definition A topological space X is reversible if every continuous bijection f : X X is a homeomorphism. Some examples of reversible spaces are: discrete spaces

5 Definition A topological space X is reversible if every continuous bijection f : X X is a homeomorphism. Some examples of reversible spaces are: discrete spaces compact spaces

6 Definition A topological space X is reversible if every continuous bijection f : X X is a homeomorphism. Some examples of reversible spaces are: discrete spaces compact spaces ω + 1 (a convergent sequence)

7 Definition A topological space X is reversible if every continuous bijection f : X X is a homeomorphism. Some examples of reversible spaces are: discrete spaces compact spaces ω + 1 (a convergent sequence) the real line

8 Definition A topological space X is reversible if every continuous bijection f : X X is a homeomorphism. Some examples of reversible spaces are: discrete spaces compact spaces ω + 1 (a convergent sequence) the real line R n for all integer n (Brouwer invariance of domain theorem)

9 Non-reversible spaces It is very easy to find spaces that are not reversible.

10 Non-reversible spaces It is very easy to find spaces that are not reversible. Define f : ω + ω ω + ω by:

11 Non-reversible spaces It is very easy to find spaces that are not reversible. Define f : ω + ω ω + ω by: for all n < ω f (n) = 2n f (ω + 2n) = ω + n f (ω + 2n + 1) = 2n + 1

12 Some history

13 Some history Reversible spaces were defined by Rajagopalan and Wilanski, 1965.

14 Some history Reversible spaces were defined by Rajagopalan and Wilanski, In the 2015 Yokohama conference Vitalij Chatyrko talked about reversible spaces ( On hereditarily reversible spaces, by Chatyrko, Han, Hattori).

15 Some history Reversible spaces were defined by Rajagopalan and Wilanski, In the 2015 Yokohama conference Vitalij Chatyrko talked about reversible spaces ( On hereditarily reversible spaces, by Chatyrko, Han, Hattori). Question (Chatyrko, Han, Hattori) Is there any compact, hereditarily reversible space different from ω + 1?

16 Some history Reversible spaces were defined by Rajagopalan and Wilanski, In the 2015 Yokohama conference Vitalij Chatyrko talked about reversible spaces ( On hereditarily reversible spaces, by Chatyrko, Han, Hattori). Question (Chatyrko, Han, Hattori) Is there any compact, hereditarily reversible space different from ω + 1? It is known that such a space has no copies of ω + ω or βω so it is an Efimov space.

17 Our problem If p ω, then ω {p} βω is reversible.

18 Our problem If p ω, then ω {p} βω is reversible. Question (Version 1) What other countable spaces with a unique non-isolated point are reversible?

19 Our problem If p ω, then ω {p} βω is reversible. Question (Version 1) What other countable spaces with a unique non-isolated point are reversible? Let F be a filter on ω (always extending the Fréchet filter).

20 Our problem If p ω, then ω {p} βω is reversible. Question (Version 1) What other countable spaces with a unique non-isolated point are reversible? Let F be a filter on ω (always extending the Fréchet filter). Define ξ(f) = ω { }, where each point of ω is isolated and the neighborhoods of are of the form F { } where F F.

21 Our problem If p ω, then ω {p} βω is reversible. Question (Version 1) What other countable spaces with a unique non-isolated point are reversible? Let F be a filter on ω (always extending the Fréchet filter). Define ξ(f) = ω { }, where each point of ω is isolated and the neighborhoods of are of the form F { } where F F. These spaces have been studied by García-Ferreira and Uzcátegi.

22 Our problem If p ω, then ω {p} βω is reversible. Question (Version 1) What other countable spaces with a unique non-isolated point are reversible? Let F be a filter on ω (always extending the Fréchet filter). Define ξ(f) = ω { }, where each point of ω is isolated and the neighborhoods of are of the form F { } where F F. These spaces have been studied by García-Ferreira and Uzcátegi. Every countable space with a unique non-isolated point is of the form ξ(f) for some F.

23 Filters and βω A filter F on ω correspond to closed subspaces K F of βω under Stone duality.

24 Filters and βω A filter F on ω correspond to closed subspaces K F of βω under Stone duality. (And K F ω since F extends the Frechét filter)

25 Filters and βω A filter F on ω correspond to closed subspaces K F of βω under Stone duality. (And K F ω since F extends the Frechét filter) Lemma Let F be a filter on ω. Then F is not reversible if and only if there is a bijection f : ω ω such that f [K F ] is a proper subset of K F.

26 When ξ(f) has convergent sequences Lemma Let F be a filter on ω. (a) ξ(f) is a convergent sequence if and only if F is the Fréchet filter if and only if K F = ω. (b) ξ(f) contains a convergent sequence if and only if int ω (K F ). (c) ξ(f) is Fréchet-Urysohn if and only if K F is a regular closed subset of ω. (d) F is an ultrafilter if and only if K F = 1.

27 When ξ(f) has convergent sequences Lemma Let F be a filter on ω. (a) ξ(f) is a convergent sequence if and only if F is the Fréchet filter if and only if K F = ω. (b) ξ(f) contains a convergent sequence if and only if int ω (K F ). (c) ξ(f) is Fréchet-Urysohn if and only if K F is a regular closed subset of ω. (d) F is an ultrafilter if and only if K F = 1. Corollary If ξ(f) is reversible and contains a convergent sequence, then ξ(f) is a convergent sequence.

28 Convergent sequences K F

29 When ξ(f) has NO convergent sequences (So K F has empty interior.)

30 When ξ(f) has NO convergent sequences (So K F has empty interior.) Question (Version 2) Given a space X embeddable in ω, is there a filter F with K F X and (a) F is reversible; (b) F is non-reversible?

31 When ξ(f) has NO convergent sequences (So K F has empty interior.) Question (Version 2) Given a space X embeddable in ω, is there a filter F with K F X and (a) F is reversible; (b) F is non-reversible? Question (Version 3) Can the filter F be a weak P-filter? or a P-filter under Martin s axiom?

32 Reversible general case Theorem There exists a filter F 0 on ω with the following properties (a) any filter that extends F 0 is reversible, (b) K F0 is nowhere dense, and (c) if X is any closed subset of ω, there exists a filter F F 0 such that K F is homeomorphic to X.

33 Reversible general case Theorem There exists a filter F 0 on ω with the following properties (a) any filter that extends F 0 is reversible, (b) K F0 is nowhere dense, and (c) if X is any closed subset of ω, there exists a filter F F 0 such that K F is homeomorphic to X. Proof. For each n < ω, let p n be a weak P-point. Assume that p n and p m have different RK types if n m.

34 Reversible general case Theorem There exists a filter F 0 on ω with the following properties (a) any filter that extends F 0 is reversible, (b) K F0 is nowhere dense, and (c) if X is any closed subset of ω, there exists a filter F F 0 such that K F is homeomorphic to X. Proof. For each n < ω, let p n be a weak P-point. Assume that p n and p m have different RK types if n m.then F 0 = {A ω ω : n < ω m m [A ({m} ω) p m ]}

35 Reversible general case

36 Non-reversible Theorem Let X be any compact, 0-dimensional, extremally disconnected space of weight c. Then there is a non-reversible filter F on ω such that K F is homeomorphic to X.

37 Non-reversible Theorem Let X be any compact, 0-dimensional, extremally disconnected space of weight c. Then there is a non-reversible filter F on ω such that K F is homeomorphic to X. Theorem Let X be a compact extremally disconnected space that can be embedded in ω as a weak P-set. Moreover, assume that there exists a proper clopen subspace of X homeomorphic to X. Then there is a non-reversible filter F on ω such that K F is a weak P-set homeomorphic to X.

38 When X has a proper clopen subspace homeomorphic to itself In this case, there are open, pairwise disjoint U n (n ω) all homeomorphic to X.

39 When X has a proper clopen subspace homeomorphic to itself In this case, there are open, pairwise disjoint U n (n ω) all homeomorphic to X

40 Counterexample There exists a compact, extremally disconnected space X that is separable (thus, embeddable in ω ) and no clopen subset of X is homeomorphic to X.

41 Counterexample There exists a compact, extremally disconnected space X that is separable (thus, embeddable in ω ) and no clopen subset of X is homeomorphic to X. Let X = β(ω <ω ), where the topology is given by a collection {p s : s ω <ω } ω RK-incomparable.

42 Counterexample There exists a compact, extremally disconnected space X that is separable (thus, embeddable in ω ) and no clopen subset of X is homeomorphic to X. Let X = β(ω <ω ), where the topology is given by a collection {p s : s ω <ω } ω RK-incomparable. U is open in ω <ω for every s U, {n < ω : s n U} p s

43 Counterexample There exists a compact, extremally disconnected space X that is separable (thus, embeddable in ω ) and no clopen subset of X is homeomorphic to X. Let X = β(ω <ω ), where the topology is given by a collection {p s : s ω <ω } ω RK-incomparable. U is open in ω <ω for every s U, {n < ω : s n U} p s If X is embedded as a weak P-set of ω, then the corresponding filter is reversible.

44 Counterexample There exists a compact, extremally disconnected space X that is separable (thus, embeddable in ω ) and no clopen subset of X is homeomorphic to X. Let X = β(ω <ω ), where the topology is given by a collection {p s : s ω <ω } ω RK-incomparable. U is open in ω <ω for every s U, {n < ω : s n U} p s If X is embedded as a weak P-set of ω, then the corresponding filter is reversible. However, there is a non-reversible filter F such that K F X by our previous result

45 Counterexample There exists a compact, extremally disconnected space X that is separable (thus, embeddable in ω ) and no clopen subset of X is homeomorphic to X. Let X = β(ω <ω ), where the topology is given by a collection {p s : s ω <ω } ω RK-incomparable. U is open in ω <ω for every s U, {n < ω : s n U} p s If X is embedded as a weak P-set of ω, then the corresponding filter is reversible. However, there is a non-reversible filter F such that K F X by our previous result (K F is not a weak P-set).

46 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X.

47 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X. Sketch of proof. Let ϕ : ω X (c + 1) c be a continuous map (Kunen-Dow).

48 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X. Sketch of proof. Let ϕ : ω X (c + 1) c be a continuous map (Kunen-Dow).We construct a decreasing sequence {K α : α c} and a decreasing sequence {I α : α < c} P(c)

49 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X. Sketch of proof. Let ϕ : ω X (c + 1) c be a continuous map (Kunen-Dow).We construct a decreasing sequence {K α : α c} and a decreasing sequence {I α : α < c} P(c) with (π Iα ϕ)[k α ] = X (c + 1) Iα.

50 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X. Sketch of proof. Let ϕ : ω X (c + 1) c be a continuous map (Kunen-Dow).We construct a decreasing sequence {K α : α c} and a decreasing sequence {I α : α < c} P(c) with (π Iα ϕ)[k α ] = X (c + 1) Iα. Let K = {K α : α < c} weak P-set,

51 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X. Sketch of proof. Let ϕ : ω X (c + 1) c be a continuous map (Kunen-Dow).We construct a decreasing sequence {K α : α c} and a decreasing sequence {I α : α < c} P(c) with (π Iα ϕ)[k α ] = X (c + 1) Iα. Let K = {K α : α < c} weak P-set, we can make (π ϕ) K : K X irreducible,

52 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X. Sketch of proof. Let ϕ : ω X (c + 1) c be a continuous map (Kunen-Dow).We construct a decreasing sequence {K α : α c} and a decreasing sequence {I α : α < c} P(c) with (π Iα ϕ)[k α ] = X (c + 1) Iα. Let K = {K α : α < c} weak P-set, we can make (π ϕ) K : K X irreducible, thus, a homeomorphism.

53 Reversible + weak P-space Theorem Let X be a compact extremally disconnected space that is a continuous image of ω. Then there is a reversible filter F such that K F is a weak P-set homeomorphic to X. Sketch of proof. Let ϕ : ω X (c + 1) c be a continuous map (Kunen-Dow).We construct a decreasing sequence {K α : α c} and a decreasing sequence {I α : α < c} P(c) with (π Iα ϕ)[k α ] = X (c + 1) Iα. Let K = {K α : α < c} weak P-set, we can make (π ϕ) K : K X irreducible, thus, a homeomorphism. We can make the filter of neighborhoods of K reversible (new part of proof).

54 P-set with MA Non-reversible P-filters can be found with the shifting technique.

55 P-set with MA Non-reversible P-filters can be found with the shifting technique. Theorem Let X be a separable, compact, extremally disconnected space. Then MA implies that there is a reversible filter F that is generated by a tower of height c such that K F is a P-set homeomorphic to X.

56 P-set with MA Non-reversible P-filters can be found with the shifting technique. Theorem Let X be a separable, compact, extremally disconnected space. Then MA implies that there is a reversible filter F that is generated by a tower of height c such that K F is a P-set homeomorphic to X. Question Is it possible to remove separability in the statement of the previous theorem?

57 Thank you.