## Kernels in dimension theory

HTML articles powered by AMS MathViewer

- by J. M. Aarts and T. Nishiura PDF
- Trans. Amer. Math. Soc.
**178**(1973), 227-240 Request permission

## Abstract:

All spaces are metrizable. A conjecture of de Groot states that a weak inductive dimension theory beginning with the class of compact spaces will characterize those spaces which can be extended to a compact space by the adjunction of a set of dimension not exceeding*n*. Nagata has proposed a variant of this conjecture as a means of finding insights into the original conjecture. (See Internat. Sympos. on Extension Theory, Berlin, 1967, pp. 157-161.) The proposed variant replaces compact with $\sigma$-compact. The present paper concerns a study of strong inductive dimension theory beginning with an arbitrary class of spaces. The study is motivated by the above two conjectures. It indicates that a theory of kernels is a more natural by-product of inductive theory than a theory of extensions. An example has resulted which, with the aid of the developed theory and the Baire category theorem, resolves the second conjecture in the negative. The original conjecture is still unresolved. It is also shown that the notion of kernels results in a further generalization of Lelek’s form of the dimension lowering map theorem (Colloq. Math. 12 (1964), 221-227. MR

**31**#716).

## References

- J. M. Aarts,
*Completeness degree. A generalization of dimension*, Fund. Math.**63**(1968), 27–41. MR**232341**, DOI 10.4064/fm-63-1-27-41 - J. M. Aarts,
*A characterization of strong inductive dimension*, Fund. Math.**70**(1971), no. 2, 147–155. MR**284989**, DOI 10.4064/fm-70-2-147-155 - R. Engelking,
*On Borel sets and $B$-measurable functions in metric spaces*, Prace Mat.**10**(1966), 145–149 (1967). MR**0209163** - Johannes de Groot,
*Topologische Studien. Compactificatie, Voortzetting van Afbeeldingen en Samenhang*, University of Groningen, 1942 (Dutch). Thesis. MR**0013299** - J. de Groot and T. Nishiura,
*Inductive compactness as a generalization of semi-compactness*, Fund. Math.**58**(1966), 201–218. MR**196704**, DOI 10.4064/fm-58-2-201-218 - Witold Hurewicz,
*Normalbereiche und Dimensionstheorie*, Math. Ann.**96**(1927), no. 1, 736–764 (German). MR**1512351**, DOI 10.1007/BF01209199
C. Kuratowski, - A. Lelek,
*Dimension and mappings of spaces with finite deficiency*, Colloq. Math.**12**(1964), 221–227. MR**176444**, DOI 10.4064/cm-12-2-221-227 - Kiiti Morita,
*On the dimension of normal spaces. II*, J. Math. Soc. Japan**2**(1950), 16–33. MR**39990**, DOI 10.2969/jmsj/00210016 - Kiiti Morita,
*Normal families and dimension theory for metric spaces*, Math. Ann.**128**(1954), 350–362. MR**65906**, DOI 10.1007/BF01360142
J. Nagata, - Togo Nishiura,
*Inductive invariants and dimension theory*, Fund. Math.**59**(1966), 243–262. MR**203692**, DOI 10.4064/fm-59-3-243-262 - Ju. M. Smirnov,
*Über die Dimension der Adjunkten bei Kompaktifizierungen*, Monatsb. Deutsch. Akad. Wiss. Berlin**7**(1965), 230–232 (German). MR**195061** - A. H. Stone,
*Absolute $F_{\sigma }$ spaces*, Proc. Amer. Math. Soc.**13**(1962), 495–499. MR**138088**, DOI 10.1090/S0002-9939-1962-0138088-4 - Gordon Thomas Whyburn,
*Analytic topology*, American Mathematical Society Colloquium Publications, Vol. XXVIII, American Mathematical Society, Providence, R.I., 1963. MR**0182943**

*Topologie*. Vol. I, 4ème éd., Monografie Mat., tom 20, PWN, Warsaw, 1958. MR

**19**, 873.

*Modern dimension theory*, Bibliotheca Math., vol. 6, Interscience, New York, 1965. MR

**34**#8380. —,

*Some aspects of extension theory in general topology*, Internat. Sympos. on Extension Theory, Berlin, 1967, pp. 157-161.

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**178**(1973), 227-240 - MSC: Primary 54F45
- DOI: https://doi.org/10.1090/S0002-9947-1973-0321037-5
- MathSciNet review: 0321037