Skip to main content Accessibility help
×
Home

Extensions of continuous functions: reflective functors

  • W. N. Hunsaker (a1) and S. A. Naimpally (a2)

Abstract

The purpose of this paper is to develop a general technique for attacking problems involving extensions of continuous functions from dense subspaces and to use it to obtain new results as well as to improve some of the known ones. The theory of structures developed by Harris is used to get some general results relating filters and covers. A necessary condition is derived for a continuous function f: XY to have a continuous extension : λx → λy where λZ denotes a given extension of the space Z. In the case of simple extensions, is continuous and in the case of strict extensions is θ-continuous. In the case of strict extensions, sufficient conditions for uniqueness of are derived. These results are then applied to several extensions considered by Banaschewski, Fomin, Kattov, Liu-Strecker, Blaszczyk-Mioduszewski, Rudölf, etc.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Extensions of continuous functions: reflective functors
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Extensions of continuous functions: reflective functors
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Extensions of continuous functions: reflective functors
      Available formats
      ×

Copyright

References

Hide All
[1]Alo, R.A. and Shapiro, H.L., “L-realcompactifications and normal bases”, J. Austral. Math. Soc. 9 (1969), 489495.
[2]Banaschewski, B., “Uber Hausdorffsch-minimale Erweiterungen von Räumen”, Arch. Math. 12 (1961), 355365.
[3]Banaschewski, B., “Extensions of topological spaces”, Canad. Math. Bull. 7 (1964), 122.
[4]Bentley, H.L. and Naimpally, S.A., “Wallman T1-compactification as epireflections”, General Topology and Appl. 4 (1974), 2941.
[5]Blaszczyk, A. and Miodusczewski, J., “On factorization of maps through τX”, Colloq. Math. 23 (1971), 4552.
[6]Chattopadhyay, K.C. and Njastad, Olav, “Quasi-regular nearness spaces and extensions of nearness-preserving maps”, Pacific J. Math. 105 (1983), 3351.
[7]Chattopadhyay, K.C. and Njastad, Olav, “Completion of merotopic spaces and extension of uniformly continuous maps”, Topology Appl. 15 (1983), 2944.
[8]Fan, K. and Gottesman, N., “On compactifications of Freudenthal and wallman”, Indag. Math. 14 (1952), 504510.
[9]Formin, S.V., “Extensions of topological spaces”, Ann. of Math. 44 (1943), 471480.
[10]Frolik, Z., “A generalization of realcompact spaces”, Czechoslovak Math. J. 13 (88) (1963), 127138.
[11]Gagrat, M.S. and Naimpally, S.A., “Proximity approach to extension problems”, Fund. Math. 71 (1971), 6376.
[12]Harris, D., “Structures in topology”, Mem. Amer. Math. Soc. 115 (1971), 196.
[13]Harris, D., “Katetov extension as a functor”, Math. Ann. 193 (1971), 171175.
[14]Harris, D., “The Wallman compactification is an epireflection”, Proc. Amer. Math. Soc. 31 (1972), 265267.
[15]Harris, D., “The Wallman compactification as a functor”, General Topology and Appl. 1 (1971), 273281.
[16]Herrlich, H., “Categorical topoloby”, General Topology and Appl. 1 (1971), 115.
[17]Herlich, H., “On the concept of reflections in general topology”, in Contrib. Extens. Theory Topol. Struct. (VEB Deutscher Verlag der Wiss. Berlin, 1969), 105114.
[18]Herrlich, H., “A concept of nearness”, General Topology and Appl. 4 (1974), 191212.
[19]Herrlich, H., “On the extendibility of continuous functions”, General Topology and Appl. 4 (1974), 213215.
[20]Herrlich, H. and Strecker, G., “H-closed spaces and reflective subcategories”, Math. Ann. 177 (1968), 302309.
[21]Hunsaker, W.N. and Naimpally, S.A., “Hausdorff compactifications as epireflections”, Canad. Math. Bull. 17 (5) (1975), 675677.
[22]Katětov, M., “On H-closed extensions of topological spaces”, Casopis Matem. Fys. 69 (1940), 3649.
[23]Liu, C.T., “The α-closure αx of a topological space X”, Proc. Amer. Math. Soc. 22 (1969), 620624.
[24]Liu, C.T. and Strecker, G.E., “Concerning almost real-compactifications”, Czechoslovak. Math. J. 22 (1972), 181190.
[25]Mioduszewski, J. and Rudolf, L., “H-closed and extremally disconnected Hausdorff spaces”, Dissertationes Math. Rozprawy Mat. 66 (1969), 152.
[26]Naimpally, S.A., “Reflective functors via nearness”. Fund. Math. 85 (1974), 245255.
[27]Naimpally, S.A. and Warrack, B.D., Proximity Spaces, (Cambridge Univ. Press, 1970).
[28]Rudolf, L., “θ-continuous extensions of maps on τX”, Fund. Math. 74 (1972), 111131.
[29]van der Slot, J., “A general realcompactification methodFund. Math. 67 (1970), 255263.
[30]Steiner, E.F., “Wallman spaces and compactifications”, Fund. Math. 61 (1968), 295304.
[31]Strecker, G.E., “Epireflection operators vs. perfect morphisms and closed classes of epimorphisms”, Bull. Austral. Math. Soc. 7 (1972), 359366.
[32]Taimanov, A.D., “On extension of continuous mappings of topological spaces”. Mat. Sb. 31 (1952), 459463.
[33]Velicko, W.V., “H-closed topological spaces”, American Mathematical Society Translations, Series 2 78 (1968), 103118.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed