Baire Sets Research Articles

On Fuzzy Baire-Separated Spaces and Related Concepts

In this paper, a new class of fuzzy topological spaces, namely fuzzy Baire-separated spaces is introduced in terms of fuzzy Baire sets. Several characterizations of fuzzy Baire-separated spaces are established. It is shown that fuzzy Baire sets lie between disjoint fuzzy P-sets and fuzzy F<sub>σ</sub>- sets in a fuzzy Baire-separated space. Conditions under which fuzzy topological spaces become fuzzy Baire-separated spaces are established. Fuzzy nowhere dense sets are fuzzy closed sets in fuzzy nodec spaces and subsequently a question will arise. Which fuzzy topological spaces [other than fuzzy hyperconnected spaces, fuzzy globally disconnected spaces] have fuzzy closed sets with fuzzy nowhere denseness? For this, fuzzy topological spaces having fuzzy closed sets with fuzzy nowhere denseness are identified in this paper. It is verified that fuzzy ultraconnected spaces are non fuzzy Baire -separated spaces. The means, by which fuzzy weakly Baire space become fuzzy Baire -separated spaces and in turn fuzzy Baire - separated spaces become fuzzy seminormal spaces, are obtained. There are scope in this paper for exploring the inter-relations between fuzzy Baire spaces and Baire -separated spaces.

The exact consistency strength of the generic absoluteness for the universally Baire sets

Abstract A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. $\mathsf {Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The $\mathsf {Largest\ Suslin\ Axiom}$ ( $\mathsf {LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let $\mathsf {LSA-over-uB}$ be the statement that in all (set) generic extensions there is a model of $\mathsf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory, $\mathsf {Sealing}$ is equiconsistent with $\mathsf {LSA-over-uB}$ . In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that $\mathsf {Sealing}$ is weaker than the theory ‘ $\mathsf {ZFC} +$ there is a Woodin cardinal which is a limit of Woodin cardinals’. A variation of $\mathsf {Sealing}$ , called $\mathsf {Tower\ Sealing}$ , is also shown to be equiconsistent with $\mathsf {Sealing}$ over the same large cardinal theory. The result is proven via Woodin’s $\mathsf {Core\ Model\ Induction}$ technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of $\mathsf {CMI}$ as explained in the paper.

THE CONSISTENCY STRENGTH OF THE PERFECT SET PROPERTY FOR UNIVERSALLY BAIRE SETS OF REALS

AbstractWe show that the statement “every universally Baire set of reals has the perfect set property” is equiconsistent modulo ZFC with the existence of a cardinal that we call virtually Shelah for supercompactness (VSS). These cardinals resemble Shelah cardinals and Shelah-for-supercompactness cardinals but are much weaker: if $0^\sharp $ exists then every Silver indiscernible is VSS in L. We also show that the statement $\operatorname {\mathrm {uB}} = {\boldsymbol {\Delta }}^1_2$ , where $\operatorname {\mathrm {uB}}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $\Sigma _2$ -reflecting VSS cardinal.

λ - Borel and λ - Baire Credibility on Normal Topological Space

In this paper, will be presented the concept of Borel sets and Baire sets on normal topological spaces and locally compact topological spaces as new types of λ – system and study the basic characteristics related to each of them and obtain results related to them.

SEALING OF THE UNIVERSALLY BAIRE SETS

AbstractA set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ ( ${\sf LSA}$ ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf over}$ - ${\sf uB}$ be the statement that in all (set) generic extensions there is a model of $\sf {LSA}$ whose Suslin, co-Suslin sets are the universally Baire sets. We outline the proof that over some mild large cardinal theory, $\sf {Sealing}$ is equiconsistent with $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ . In fact, we isolate an exact theory (in the hierarchy of strategy mice) that is equiconsistent with both (see Definition 3.1). As a consequence, we obtain that $\sf {Sealing}$ is weaker than the theory “ $\sf {ZFC}$ + there is a Woodin cardinal which is a limit of Woodin cardinals.” This significantly improves upon the earlier consistency proof of $\sf {Sealing}$ by Woodin. A variation of $\sf {Sealing}$ , called $\sf {Tower \ Sealing}$ , is also shown to be equiconsistent with $\sf {Sealing}$ over the same large cardinal theory. We also outline the proof that if V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, then $\sf {Sealing}$ holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to the aforementioned equiconsistency result, where it is shown that $\sf {Sealing}$ holds in a generic extension of a certain minimal universe. This theorem is more general in that no minimal assumption is needed. A corollary of this is that $\sf {LSA}$ - $\sf {over}$ - $\sf {uB}$ is not equivalent to $\sf {Sealing}$ .

𝖲𝖾𝖺𝗅𝗂𝗇𝗀 from iterability

We show that if V V has a proper class of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy (as defined in the paper) then S e a l i n g \mathsf {Sealing} holds after collapsing the successor of the least strong cardinal to be countable. This result is complementary to other work by the authors where it is shown that S e a l i n g \mathsf {Sealing} holds in a generic extension of a certain minimal universe. The current theorem is more general in that no minimality assumption is needed. A corollary of the main theorem is that S e a l i n g \mathsf {Sealing} is consistent relative to the existence of a Woodin cardinal which is a limit of Woodin cardinals. This improves significantly on the first consistency of S e a l i n g \mathsf {Sealing} obtained by W.H. Woodin. The L a r g e s t S u s l i n A x i o m \mathsf {Largest\ Suslin\ Axiom} ( L S A \mathsf {LSA} ) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let L S A - o v e r - u B \mathsf {LSA\text {-}over\text {-}uB} be the statement that in all (set) generic extensions there is a model of L S A \mathsf {LSA} whose Suslin, co-Suslin sets are the universally Baire sets. The other main result of the paper shows that assuming V V has a proper class of inaccessible cardinals which are limit of Woodin cardinals, a strong cardinal, and a generically universally Baire iteration strategy, in the universe V [ g ] V[g] , where g g is V V -generic for the collapse of the successor of the least strong cardinal to be countable, the theory L S A \mathsf {LSA} - o v e r \mathsf {over} - U B \mathsf {UB} fails; this implies that L S A - o v e r - U B \mathsf {LSA\text {-}over\text {-}UB} is not equivalent to S e a l i n g \mathsf {Sealing} (over the base theory of V [ g ] V[g] ). This is interesting and somewhat unexpected, in light of other work by the authors. Compare this result with Steel’s well-known theorem that “ A D L ( R ) \mathsf {AD}^{L(\mathbb {R})} holds in all generic extensions” is equivalent to “the theory of L ( R ) L(\mathbb {R}) is sealed” in the presence of a proper class of measurable cardinals.

Completely Baire spaces, Menger spaces, and projective sets

W. Hurewicz proved that analytic Menger sets of reals are σ-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by V=L for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgöz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with ωth power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.

Topological Ramsey spaces and metrically Baire sets

We characterize a class of topological Ramsey spaces such that each element R of the class induces a collection {Rk}k<ω of projected spaces which have the property that every Baire set is Ramsey. Every projected space Rk is a subspace of the corresponding space of length-k approximation sequences with the Tychonoff, equivalently metric, topology. This answers a question of S. Todorcevic and generalizes some results of Carlson, Carlson–Simpson, Prömel–Voigt, and Voigt. We also present a new family of topological Ramsey spaces contained in the aforementioned class which generalize the spaces of ascending parameter words of Carlson–Simpson and Prömel–Voigt and the spaces FINm[∞], 0<m<ω, of block sequences defined by Todorcevic.

Bounded Martin’s Maximum with an Asterisk

We isolate natural strengthenings of Bounded Martin’s Maximum which we call BMM∗BMM∗ and A−BMM∗,++A−BMM∗,++ (where AA is a universally Baire set of reals), and we investigate their consequences. We also show that if A−BMM∗,++A−BMM∗,++ holds true for every set of reals AA in L(R)L(R), then Woodin’s axiom (∗)(∗) holds true. We conjecture that MM++MM++ implies A−BMM∗,++A−BMM∗,++ for every AA which is universally Baire.

Strongly dominating sets of reals

We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573---1581, 1995). We prove that for every $${\kappa < \mathfrak{b}}$$ ? < b a $${\kappa}$$ ? -Suslin set $${A\subseteq{}^\omega\omega}$$ A ⊆ ? ? is strongly dominating if and only if A has a Laver perfect subset. We also investigate the structure of the class l of Baire sets for the Laver category base and compare the ?-ideal of sets which are not strongly dominating with the Laver ideal l 0.

Monomorphisms in spaces with Lindelöf filters

SpFi is the category of spaces with filters: an object is a pair (X, ℱ), X a compact Hausdorff space and ℱ a filter of dense open subsets of X. A morphism f: (Y, Open image in new window ) → (X, ℱ) is a continuous function f: Y → X for which f−1(F) ∈ Open image in new window whenever F ∈ ℱ. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shallspecifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in SpFi. Of course, these monomorphisms need not be one-to-one. For general SpFi we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category LSpFi of spaces with Lindelof filters, meaning filters with a base of Lindelof, or cozero, sets, the criterion becomes a real characterization with several foci (C(X), Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given (X, ℱ) ∈ LSpFi. Considerable attempt is made to keep the discussion “topological, ” i.e., within SpFi, and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean l-groups with unit, roughly dual to monics in LSpFi.

Universally Baire sets and definable well-orderings of the reals

AbstractLet n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n − 2 strong cardinals) that every Σ1n-set of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses “David's trick” in the presence of inner models with strong cardinals.

α-Specker spaces

This paper is motivated by work on Specker spaces and a recent article of the authors on the ring of α-quotients. Here α denotes an uncountable regular cardinal or else ∞, indicating no cardinal constraint whatsoever. All spaces are compact Hausdorff, and for the most part zero-dimensional. Three strains of the “ α-Specker” condition are studied: strong, weak, and one in between which is not qualified. One of the main results characterizes these conditions, for each α and each space X, in terms of the containment of C( X) in the ring of α-quotients of S( K α X), where the latter denotes the algebra of continuous functions with finite range, defined on an appropriate cover K α X of X. “Weakly c + -Specker” is equivalent to “Specker”. The paper examines the ω 1-Specker conditions, proving that “weakly ω 1-Specker” and “ ω 1-Specker” are equivalent. In fact, it is shown that X is weakly ω 1-Specker if and only if for each f∈ C( X) there is a Baire set B with meagre complement such that f( B) is countable.

The Range of Vector Measures over Topological Sets and a Unified Liapunov Theorem

Let (X, Σ, μ) be a finite, nonatomic, measure space. Let G=span{g1, g2, …, gn}⊆L1, and let the support of G be X, a.e. For f∈L∞, put M(f)=(∫Xfg1dμ, ∫Xfg2dμ, …, ∫Xfgndμ). Then Q={M(f):f∈B(L∞)} is a compact convex set. Liapunov's classsical theorem is that also Q={M(s):s∈extB(L∞)}. This paper characterizes when the functions, s, in Liapunov's theorem can further be restricted to being the signs of continuous functions. That is, suppose there is a topology on X, and that Σ is the Baire sets. Let S be the collection of supports of continuous non-negative functions. Theorem.The following are equivalent: (i)Q={M(s):s∈extB(L∞),ands=sgnf for some f∈C(X)}, (ii)for every g∈G, {x:g(x)>0}and{x:g(x)⩽0} are in S, a.e. The theorem is proved in a general setting. If the σ-field in the general theorem is arbitrary, the theorem becomes Liapunov's Theorem. The theorem above results when the σ-field is the Baire sets. A setting with the Borel sets produces Q as the range of M over extreme functions that are both lower semi-continuous a.e. and upper semi-continuous a.e.

Spectral Properties of Abelian C*-Algebras

Let \({\mathcal{A}}\) be an Abelian unital C*-algebra and let \(\hat {\mathcal{A}}\) denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of \({\mathcal{A}}\) to be unitarily equivalent to a representation in which the elements of \({\mathcal{A}}\) act multiplicatively, by their Gelfand transforms, on a space L2(\(\hat {\mathcal{A}}\),μ), where μ is a positive measure on the Baire sets of \(\hat {\mathcal{A}}\). We also compare these conditions with the multiplicity-free property of a representation.

A minimal counterexample to universal baireness

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

Additive and Multiplicative Ramsey Theory in the Reals and the Rationals

Let a finite partition F of the real interval (0,1) be given. We show that if every member of F is measurable or if every member of F is a Baire set, then one member of F must contain a sequence with all of its finite sums and products (and, in the measurable case, all of its infinite sums as well). These results are obtained by using the algebraic structure of the Stone–Čech compactification of the real numbers with the discrete topology. They are also obtained by elementary methods. In each case we in fact get significant strengthenings of the above stated results (with different strengthenings obtained by the algebraic and elementary methods). Some related (although weaker) results are established for arbitrary partitions of the rationals and the dyadic rationals, and a counterexample is given to show that even weak versions of the combined additive and multiplicative results do not hold in the dyadic rationals.

Characterizations of weakly compact operators on $C_o(T)$

Let T be a locally compact Hausdorff space and let Co(T) ={f: T -* C, f is continuous and vanishes at infinity} be provided with the supremum norm. Let l3B(T) and 13o(T) be the a-rings generated by the compact subsets and by the compact G6 subsets of T, respectively. The members of Bc (T) are called o-Borel sets of T since they are precisely the or-bounded Borel sets of T. The members of 130(T) are called the Baire sets of T. M(T) denotes the dual of Co(T). Let X be a quasicomplete locally convex Hausdorff space. Suppose u: Co(T) -* X is a continuous linear operator. Using the Baire and a-Borel characterizations of weakly compact sets in M(T) as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator u to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of a-additive X-valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.

Refinements of the density and ℐ-density topologies

Given an arbitrary ideal J \mathcal {J} on the real numbers, two topologies are defined that are both finer than the ordinary topology. There are nonmeasurable, non-Baire sets that are open in all of these topologies, independent of J \mathcal {J} . This shows why the restriction to Baire sets is necessary in the usual definition of the J \mathcal {J} -density topology. It appears to be difficult to find such restrictions in the case of an arbitrary ideal.

Finite group actions on the moduli space of self-dual connections. I

Let M M be a smooth simply connected closed 4 4 -manifold with positive definite intersection form. Suppose a finite group G G acts smoothly on M M . Let π : E → M \pi :E \to M be the instanton number one quaternion line bundle over M M with a smooth G G -action such that π \pi is an equivariant map. We first show that there exists a Baire set in the G G -invariant metrics on M M such that the moduli space M ∗ G \mathcal {M}_ * ^G of G G -invariant irreducible self-dual connections is a manifold. By utilizing the G G -transversality theory of T \text {T} . Petrie, we then identify cohomology obstructions to globally perturbing the full space M ∗ {\mathcal {M}_ * } of irreducible self-dual connections to a G G -manifold when G = Z 2 G = {{\mathbf {Z}}_2} and the fixed point set of the Z 2 {\mathbf {Z}}_2 action on M M is a nonempty collection of isolated points and Riemann surfaces.