A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bilimits of topoi, we show that every Galois topos has a point. introduction. Galois topoi (definition 1.5) arise in Grothe...

The degree of a structure si is the Turing degree of its open diagram £»(j/) , coded as a subset of a>. Implicit in the definition is a particular presentation of the structure; the degree is not an isomorphism invariant. We prove that if a Boolean algebra si has a copy of low 2 degree, then there is a recursive Boolean algebra 33 which is isomorphic to si . This builds on work of Downey and Jo...

As is well known, Lovfisz Local Lemma implies that every d-uniform d-regular hyper-graph is 2-colorable, provided d > 9. We present a different proof of a slightly stronger result; every d-uniform d-regular hypergraph is 2-colorable, provided d > 8.

In [16, Theorem 3.12, p. 397], Ungar answered a question of Burgess [2] by showing that every 2-homogeneous metric continuum is locally connected. A short, elementary and elegant proof of this result has been given by Whittington [17] who has omitted a powerful result of Effros (viz. Theorem 2.1 of [4, p. 39]) used by Ungar. It is observed in this note that Whittington's proof can be applied to...