On classification of sigma hereditary disconnected spaces

T.M.Radul

We introduce and investigate a transfinite dimension function ${\rm p \,}$ with the property: ${\rm p \,}(X)=0$ iff $X$ is a punctiform space. We prove that the dimension ${\rm p \,}$ classify sigma hereditary disconnected spaces. Using this fact we show that each complete sigma hereditary disconnected space is a C-space in the sense of Haver.

1. Steinke G. A new dimension by means of separating sets, Arch.Math 40 (1983), 273--282.

2. Arenas F.G., Chatyrko V.A., Puertas M.L. Transfinite extension of Steinke's dimension, Acta Math. Hung. 88 (2000), no. 1--2, 105--112.

3. Engelking R. Dimension Theory, PWN, Warszawa, 1978.

4. Hurewicz W. Ueber unendlich-dimensionale Punktmengen, Proc.Acad.Amsterd., 31 (1928), 916--922.

5. Krasinkiewicz J., Essential mappings onto products of manifolds, Geometric and Algebraic Topology, 18. Banach Center Publ. PWN, 1986, 377--406.

6. Banakh T., Cauty R. On universality of countable and weak products of sigma hereditarily disconnected spaces, Fund.Math. 167 (2001), no. 2, 97--109.

7. Addis D.F., Gresham J.H. A class of infinite dimensional spaces. Part 1:Dimension theory and Alexandroff's problem, Fund. Math. 101 (1978), 195--205.

8. Haver W.E. A covering property for metric spaces. Lecture Notes in Math. 375 (1974), 108--113.

9. Borst P. Some remarks concerning C-spaces, preprint.

10. Nagata J. On the countable sum of zero-dimensional metric spaces Fund. Math. 48 (1960).

11. Pol R. A weakly infinite-dimensional compactum which is not countable-dimensional, Proc. Amer. Math. Soc. 82 (1981), 634--636.

	On classification of sigma hereditary disconnected spaces
Author	T.M.Radul tarasradul@yahoo.co.uk Lviv Ivan Franko National University
Abstract	We introduce and investigate a transfinite dimension function ${\rm p \,}$ with the property: ${\rm p \,}(X)=0$ iff $X$ is a punctiform space. We prove that the dimension ${\rm p \,}$ classify sigma hereditary disconnected spaces. Using this fact we show that each complete sigma hereditary disconnected space is a C-space in the sense of Haver.
Keywords	transfinite dimension function, punctiform space, sigma hereditary disconnected space
DOI	doi:10.30970/ms.26.1.97-100
Reference	1. Steinke G. A new dimension by means of separating sets, Arch.Math 40 (1983), 273--282. 2. Arenas F.G., Chatyrko V.A., Puertas M.L. Transfinite extension of Steinke's dimension, Acta Math. Hung. 88 (2000), no. 1--2, 105--112. 3. Engelking R. Dimension Theory, PWN, Warszawa, 1978. 4. Hurewicz W. Ueber unendlich-dimensionale Punktmengen, Proc.Acad.Amsterd., 31 (1928), 916--922. 5. Krasinkiewicz J., Essential mappings onto products of manifolds, Geometric and Algebraic Topology, 18. Banach Center Publ. PWN, 1986, 377--406. 6. Banakh T., Cauty R. On universality of countable and weak products of sigma hereditarily disconnected spaces, Fund.Math. 167 (2001), no. 2, 97--109. 7. Addis D.F., Gresham J.H. A class of infinite dimensional spaces. Part 1:Dimension theory and Alexandroff's problem, Fund. Math. 101 (1978), 195--205. 8. Haver W.E. A covering property for metric spaces. Lecture Notes in Math. 375 (1974), 108--113. 9. Borst P. Some remarks concerning C-spaces, preprint. 10. Nagata J. On the countable sum of zero-dimensional metric spaces Fund. Math. 48 (1960). 11. Pol R. A weakly infinite-dimensional compactum which is not countable-dimensional, Proc. Amer. Math. Soc. 82 (1981), 634--636.
Pages	97-100
Volume	26
Issue	1
Year	2006
Journal	Matematychni Studii
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