Abstract. We continue the study of (strictly) o-bounded topological groups initiated by the rst listed author and solve two problems posed earlier. It is shown here that the product of a Comfort-like topological group by a (strictly) o-bounded group is (strictly) o-bounded. Some non-trivial examples of strictly obounded free topological groups are given. We also show that o-boundedness is not productive, and strict o-boundedness cannot be characterized by means of second countable continuous homomorphic images.2000 AMS Classi cation: Primary 54H11, 22A05; Secondary 22D05, 54C50
Abstract. It is shown under CH that there exists a countably compact topological semigroup with two-sided cancellation which is not a topological group. "Wallace's question" of 40 years standing is thus settled in the negative unless CH is explicitly denied. The example is a topological subsemigroup of an uncountable product of circle groups.
Halley's method is a famous iteration for solving nonlinear equations. Some Kantorovichlike theorems have been given. The purpose of this note is to relax the region conditions and give another Kantorovich-like theorem for operator equations.
Compact, connected, totally ordered, (Hausdorff) topological groupoids, with restrictions on their sets of idempotents and with varying degrees of power associativity assumed, are examined. The paper evolves from the author's example of such a groupoid which has only two idempotents (a zero for least element, and an identity for greatest element), a compact neighborhood of the greatest element consisting of power associative elements, and which is not isomorphic to either the real thread or the nil thread. Another example given has a zero for least element, an idempotent for greatest element, and no other idempotents, and has a compact neighborhood of the greatest element consisting of an associative subgroupoid in which all products are equal to the greatest element. Theorems are given which show that these examples, and one other, in some sense, exhaust the possibilities.
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