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Short proofs of two theorems in topology

Mohammad Ismail, Andrzej Szymański (1993)

Commentationes Mathematicae Universitatis Carolinae

We present short and elementary proofs of the following two known theorems in General Topology: (i) [H. Wicke and J. Worrell] A T 1 weakly δ θ -refinable countably compact space is compact. (ii) [A. Ostaszewski] A compact Hausdorff space which is a countable union of metrizable spaces is sequential.

Sierpiński's hierarchy and locally Lipschitz functions

Michał Morayne (1995)

Fundamenta Mathematicae

Let Z be an uncountable Polish space. It is a classical result that if I ⊆ ℝ is any interval (proper or not), f: I → ℝ and α < ω 1 then f ○ g ∈ B α ( Z ) for every g ∈ B α ( Z ) ∩ Z I if and only if f is continuous on I, where B α ( Z ) stands for the αth class in Baire’s classification of Borel measurable functions. We shall prove that for the classes S α ( Z ) ( α > 0 ) in Sierpiński’s classification of Borel measurable functions the analogous result holds where the condition that f is continuous is replaced by the condition that f is locally Lipschitz...

Sigma-finiteness and regularity of generalized Radon measures.

J. Fernández Novoa (1990)

Collectanea Mathematica

We extend some known sigma-finiteness and regularity results for (locally finite) Radon measures to locally sigma-finite or locally moderated Radon measures of type (H), and we obtain other new ones. The main result states that the regularity and the sigma-finiteness are equivalent for alllocally moderated, diffused, Radon measures of type (H) in a T1 topological space which is either weakly metacompact or paralindelöf (resp. metalindelöf) and has a concassage of Lindelöf (resp. separable) subsets....

Simple motions

Athanossios Tzouvaras (1989)

Commentationes Mathematicae Universitatis Carolinae

Simplicity of Neretin's group of spheromorphisms

Christophe Kapoudjian (1999)

Annales de l'institut Fourier

Denote by 𝒯 n , n ≥ 2 , the regular tree whose vertices have valence n + 1 , ∂ 𝒯 n its boundary. Yu. A. Neretin has proposed a group N n of transformations of ∂ 𝒯 n , thought of as a combinatorial analogue of the diffeomorphism group of the circle. We show that N n is generated by two groups: the group Aut ( 𝒯 n ) of tree automorphisms, and a Higman-Thompson group G n . We prove the simplicity of N n and of a family of its subgroups.

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