We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall’s condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theore...

Let L be an Archimedean Riesz space with a weak order unit u. A sufficient condition under which Dedekind [σ-]completeness of the principal ideal Au can be lifted to L is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of C(X)-spaces. Similar results are obtained for the Riesz spaces Bn(T ), n = 1, 2, . . . , of all functions o...

This paper introduces the concepts of fuzzy pseudo-open sets in fuzzy topological spaces and fuzzy pseudo-continuous functions between fuzzy topological spaces. Several characterizations of fuzzy pseudo-open sets are established. A conditions under which fuzzy hyper-connected spaces become fuzzy Baire spaces, is obtained by means of fuzzy pseudo-open sets. Several examples are given to illustra...

The following discusses a generalization of Baire category in a continuous set. The objective is to provide a meaningful classification of subsets of a continuous set as “large” or “small” sets in linearly ordered continuous sets. In particular, for cardinal number κ, the continuous ordered set 2∗ a subset of the set of dyadic sequences of length κ is discussed. We establish that this space, an...

The study of typical continuous functions has been one of the most popular topics in classical real analysis. Several people have also investigated typical behaviour in other families of functions, such as those of bounded functions of Baire class 1, of bounded Darboux functions of Baire class 1, and of bounded derivatives, where the topology is given by the supremum norm. Kostyrko and Šalát [2...

In [8] we have considered a wide class of “well-behaved” reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [8] allowing to deal with a larger class of notions of reduction (including, among others, the Baire clas...

Let κ be an uncountable cardinal with κ = κ. In this paper we introduce Rκ, a Cauchy-complete real closed field of cardinality 2. We will prove that Rκ shares many features with R which have a key role in real analysis and computable analysis. In particular, we will prove that the Intermediate Value Theorem holds for a non-trivial subclass of continuous functions over Rκ. We propose Rκ as a can...

We study reflection principles involving nonmeager sets and the Baire Property which are consequences of the generic supercompactness of ω2, such as the principle asserting that any point countable Baire space has a stationary set of closed subspaces of weight ω1 which are also Baire spaces. These principles entail the analogous principles of stationary reflection but are incompatible with forc...