Decomposition, Mapping, and Sum Theorems of ω-Paracompact Topological Spaces

As a weaker form of ω-paracompactness, the notion of σ-ω-paracompactness is introduced. Furthermore, as a weaker form of σ-ω-paracompactness, the notion of feebly ω-paracompactness is introduced. It is proven hereinthat locally countable topological spaces are feebly ω-paracompact. Furthermore, it is proven hereinthat countably ω-paracompact σ-ω-paracompact topological spaces are ω-paracompact. Furthermore, it is proven hereinthat σ-ω-paracompactness is inverse invariant under perfect mappings with countable fibers, and as a result, is proven hereinthat ω-paracompactness is inverse invariant under perfect mappings with countable fibers. Furthermore, if A is a locally finite closed covering of a topological space X,τ with each A∈A being ω-paracompact and normal, then X,τ is ω-paracompact and normal, and as a corollary, a sum theorem for ω-paracompact normal topological spaces follows. Moreover, three open questions are raised.

Soft Semi ω-Open Sets

In this paper, we introduce the class of soft semi ω-open sets of a soft topological space (X,τ,A), using soft ω-open sets. We show that the class of soft semi ω-open sets contains both the soft topology τω and the class of soft semi-open sets. Additionally, we define soft semi ω-closed sets as the class of soft complements of soft semi ω-open sets. We present here a study of the properties of soft semi ω-open sets, especially in (X,τ,A) and (X,τω,A). In particular, we prove that the class of soft semi ω-open sets is closed under arbitrary soft union but not closed under finite soft intersections; we also study the correspondence between the soft topology of soft semi ω-open sets of a soft topological space and their generated topological spaces and vice versa. In addition to these, we introduce the soft semi ω-interior and soft semi ω-closure operators via soft semi ω-open and soft semi ω-closed sets. We prove several equations regarding these two new soft operators. In particular, we prove that these operators can be calculated using other usual soft operators in both of (X,τ,A) and (X,τω,A), and some equations focus on soft anti-locally countable soft topological spaces.

Results on Martin’s Conjecture

Martin’s conjecture is an attempt to classify the behavior of all definable functions on the Turing degrees under strong set theoretic hypotheses. Very roughly it says that every such function is either eventually constant, eventually equal to the identity function or eventually equal to a transfinite iterate of the Turing jump. It is typically divided into two parts: the first part states that every function is either eventually constant or eventually above the identity function and the second part states that every function which is above the identity is eventually equal to a transfinite iterate of the jump. If true, it would provide an explanation for the unique role of the Turing jump in computability theory and rule out many types of constructions on the Turing degrees.In this thesis, we will introduce a few tools which we use to prove several cases of Martin’s conjecture. It turns out that both these tools and these results on Martin’s conjecture have some interesting consequences both for Martin’s conjecture and for a few related topics.The main tool that we introduce is a basis theorem for perfect sets, improving a theorem due to Groszek and Slaman. We also introduce a general framework for proving certain special cases of Martin’s conjecture which unifies a few pre-existing proofs. We will use these tools to prove three main results about Martin’s conjecture: that it holds for regressive functions on the hyperarithmetic degrees (answering a question of Slaman and Steel), that part 1 holds for order preserving functions on the Turing degrees, and that part 1 holds for a class of functions that we introduce, called measure preserving functions.This last result has several interesting consequences for the study of Martin’s conjecture. In particular, it shows that part 1 of Martin’s conjecture is equivalent to a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. This suggests several possible strategies for working on part 1 of Martin’s conjecture, which we will discuss.The basis theorem that we use to prove these results also has some applications outside of Martin’s conjecture. We will use it to prove a few theorems related to Sacks’ question about whether it is provable in $\mathsf {ZFC}$ that every locally countable partial order of size continuum embeds into the Turing degrees. We will show that in a certain extension of $\mathsf {ZF}$ (which is incompatible with $\mathsf {ZFC}$ ), this holds for all partial orders of height two, but not for all partial orders of height three. Our proof also yields an analogous result for Borel partial orders and Borel embeddings in $\mathsf {ZF}$ , which shows that the Borel version of Sacks’ question has a negative answer.We will end the thesis with a list of open questions related to Martin’s conjecture, which we hope will stimulate further research.Abstract prepared by Patrick Lutz.E-mail: [email protected]

On some types of functions and a form of compactness via $ \omega _{s} $-open sets

<abstract><p>In this paper, $ \omega _{s} $-irresoluteness as a strong form of $ \omega _{s} $-continuity is introduced. It is proved that $ \omega _{s} $-irresoluteness is independent of each of continuity and irresoluteness. Also, $ \omega _{s} $-openness which lies strictly between openness and semi-openness is defined. Sufficient conditions for the equivalence between $ \omega _{s} $-openness and openness, and between $ \omega _{s} $-openness and semi-openness are given. Moreover, pre-$ \omega _{s} $ -openness which is a strong form of $ \omega _{s} $-openness and independent of each of openness and pre-semi-openness is introduced. Furthermore, slight $ \omega _{s} $-continuity as a new class of functions which lies between slight continuity and slight semi-continuity is introduced. Several results related to slight $ \omega _{s} $-continuity are introduced, in particular, sufficient conditions for the equivalence between slight $ \omega _{s} $ -continuity and slight continuity, and between slight $ \omega _{s} $ -continuity and slight semi-continuity are given. In addition to these, $ \omega _{s} $-compactness as a new class of topological spaces that lies strictly between compactness and semi-compactness is introduced. It is proved that locally countable compact topological spaces are $ \omega _{s} $ -compact. Also, it is proved that anti-locally countable $ \omega _{s} $ -compact topological spaces are semi-compact. Several implications, examples, counter-examples, characterizations, and mapping theorems are introduced related to the above concepts are introduced.</p></abstract>

JUMP OPERATIONS FOR BOREL GRAPHS

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

MEASURABLE PERFECT MATCHINGS FOR ACYCLIC LOCALLY COUNTABLE BOREL GRAPHS

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating μ-hyperfinite equivalence relations admit μ-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.