Spaces of σ-finite linear measure

Ihor Stasyuk, and Edward D. Tymchatyn. "Spaces of σ-finite linear measure." Colloquium Mathematicae 133.2 (2013): 245-252. <http://eudml.org/doc/283626>.

@article{IhorStasyuk2013,
abstract = {Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear measure then X admits a σ-finite linear measure. We answer in the strongest possible way a 1989 question (private communication) of Mauldin. We prove that if a separable metric space is a countable union of closed subspaces each of which admits finite linear measure then it admits σ-finite linear measure. In particular, it can be embedded in the 1-dimensional Nöbeling space ν₁³ so that the image has σ-finite linear measure with respect to the usual metric on ν₁³.},
author = {Ihor Stasyuk, Edward D. Tymchatyn},
journal = {Colloquium Mathematicae},
keywords = {linear Hausdorff measure; totally regular space; space of -finite linear measure; 1-dimensional Nöbeling space; Z-set},
language = {eng},
number = {2},
pages = {245-252},
title = {Spaces of σ-finite linear measure},
url = {http://eudml.org/doc/283626},
volume = {133},
year = {2013},
}

TY - JOUR
AU - Ihor Stasyuk
AU - Edward D. Tymchatyn
TI - Spaces of σ-finite linear measure
JO - Colloquium Mathematicae
PY - 2013
VL - 133
IS - 2
SP - 245
EP - 252
AB - Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear measure then X admits a σ-finite linear measure. We answer in the strongest possible way a 1989 question (private communication) of Mauldin. We prove that if a separable metric space is a countable union of closed subspaces each of which admits finite linear measure then it admits σ-finite linear measure. In particular, it can be embedded in the 1-dimensional Nöbeling space ν₁³ so that the image has σ-finite linear measure with respect to the usual metric on ν₁³.
LA - eng
KW - linear Hausdorff measure; totally regular space; space of -finite linear measure; 1-dimensional Nöbeling space; Z-set
UR - http://eudml.org/doc/283626
ER -