Hostname: page-component-6d856f89d9-76ns8 Total loading time: 0 Render date: 2024-07-16T08:47:33.154Z Has data issue: false hasContentIssue false
  • English
  • Français

A unified approach to continuous and certain non-continuous functions II

Published online by Cambridge University Press:  17 April 2009

J.K. Kohli
J.K. Kohli
Affiliation:
Department of Mathematics, Hindu College, University of Delhi, Delhi - 110007, India
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A unified theory of continuous and certain non-continuous functions, initiated in an earlier paper, is further elaborated. The proposed theory provides a common platform for dealing simultaneously with continuous functions and a host of non-continuous functions including lower (upper) semicontinuous functions, almost continuous functions, weakly continuous functions (encountered in functional analysis), c-continuous functions, δ-continuous functions, semiconnected functions, H-continuous functions s-continuous functions, ε-continuous functions of Klee and several other variants of continuity.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 1 , February 1990 , pp. 57 - 74
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Beer, Gerald, ‘Lattice semicontinuous mappings and their applications’, Houston J. Math. 13 (1987), 303–318.Google Scholar
[2]Brandenburg, H., ‘On a class of nearness spaces and the epireflective hull of developable topological space’, Proc. Internat. Topological Symp. (Belgrade 1977).Google Scholar
[3]Brandenburg, H., ‘On spaces with a G δ-basis’, Arch. Math. 35 (1980), 544–547.Google Scholar
[4]Carnahan, D., Some properties related to compactness in topological spaces (Ph.D Thesis, Univ. of Arkansas, 1973).Google Scholar
[5]Carnahan, D., ‘Locally nearly compact spaces’, Boll. Un. Mat. Ital. 6 (1972), 146–153.Google Scholar
[6]Cammaroto, F. and Lo Faro, G., ‘Sulle funzioni γ-continue’, Le Matematiche 35 (1980), 1–17.Google Scholar
[7]Commaroto, F. and Noiri, T., ‘On WC-continuous functions’, J. Korean Math. Soc. 24 (1987), 11–19.Google Scholar
[8]Franklin, S.P., ‘Spaces in which sequences suffice’, Fund. Math. 57 (1965), 107–115.CrossRefGoogle Scholar
[9]Gauld, D.B., ‘Topologies related to notions of near continuity’, Kyungpook Math. J. 21 (1981), 195–204.Google Scholar
[10]Gauld, D.B., Mrsevic, M., Reilly, I.L. and Vamanamurthy, M. K., ‘Continuity properties of functions’, Colloquia Math. Soc. Janos Bolyai 41: Topology and its Applications, Eger (1983), pp. 311–322.Google Scholar
[11]Gauld, D.B., Mrsevic, M., Reilly, I.L. and Vamanamurthy, M. K., ‘Colindlöf topologies and ℓ-continuous functions’, Glas. Mat. 19(39) (1984), 297–308.Google Scholar
[12]Gentry, Karl R. and Hoyle, Hughes B., ‘c-continuous functions’, Yokohama Math. J. 18 (1970), 71–76.Google Scholar
[13]Heldermann, N.C., ‘The category of D-completely regular spaces is simple’, Trans. Amer. Math. Soc. 262 (1980), 437–446.Google Scholar
[14]Heldermann, N.C., ‘Developability and some new regularity axioms’, Canad. J. Math. 33 (1981), 641–663.CrossRefGoogle Scholar
[15]Hewitt, E., ‘Rings of real-valued continuous functions I’, Trans. Amer. Math. Soc. 64 (1948), 45–99.Google Scholar
[16]Jones, John Jr, ‘On semiconnected mappings of topological spaces’, Proc. Amer. Math. Soc. 19 (1968), 174–175.CrossRefGoogle Scholar
[17]Klee, Victor, ‘Stability of the fixed point property’, Colloq. Math. 8 (1961), 43–46.Google Scholar
[18]Klee, Victor and Yandl, Andre, ‘Some proximate concepts in topology’, Sympos. Math. 16 (1975), 21–39.Google Scholar
[19]Kohli, J.K., ‘A unified approach to continuous and certain non-continuous functions’, Symposium on Gen. Topology and its Applications, University of Delhi, Delhi (1978). J. Austra. Math. Soc. (to appear).Google Scholar
[20]Kohli, J.K., ‘A class of mappings containing all continuous and all semiconnected mappings’, Proc. Amer. Math. Soc. 72 (1978), 175–181.CrossRefGoogle Scholar
[21]Kohli, J.K., ‘S-continuous functions, certain weak forms of regularity and complete regularity’, Math. Nachr 97 (1980), 189–196.Google Scholar
[22]Kohli, J.K., ‘S-continuous functions, certain weak forms of normality and strongly semilocally connected spaces’, Math. Nachr 99 (1980), 69–76.CrossRefGoogle Scholar
[23]Kohli, J.K., ‘A class of mappings containing all continuous mappings’, Glas. Mat. 16(36) (1981), 361–368.Google Scholar
[24]Kohli, J.K., ‘A unified view of (complete) regularity and certain variants of (complete) regularity’, Canad. J. Math. 36 (1984), 783–794.CrossRefGoogle Scholar
[25]Kohli, J.K., ‘D-continuous functions, D-Hausdorff spaces and D-regular spaces’, (submitted).Google Scholar
[26]Kohli, J.K., ‘On certain generalized notions of continuity’, (preprint).Google Scholar
[27]Levine, Norman, ‘A decomposition of continuity in topological spaces’, Amer. Math. Monthly 68 (1961), 44–46.Google Scholar
[28]Long, Paul E., ‘Concerning semiconnected mappings’, Proc. Amer. Math. Soc. 21 (1969), 117–118.Google Scholar
[29]Long, Paul E. and Carnahan, D. A., ‘Comparing almost continuous functions’, Proc. Amer. Math. Soc. 38 (1973), 413–418.CrossRefGoogle Scholar
[30]Long, Paul E. and Hamlett, T.R., ‘H-continuous functions’, Boll. Un. Mat. Ital 11 (1975), 552–558.Google Scholar
[31]Long, Paul E. and Hendrix, Michael D., ‘Properties of c-continuous functions’, Yokohama Math. J. 22 (1974), 117–125.Google Scholar
[32]Long, Paul E. and Herrington, Larry L., ‘Properties of almost continuous functions’, Boll. Un. Mat. Ital 10 (1974), 336–342.Google Scholar
[33]Long, Paul E. and Herrington, Larry L., ‘Properties of c-continuous and c-continuous functions’, Kyungpook Math. J. 15 (1975), 213–221.Google Scholar
[34]Long, Paul E. and Herrington, Larry L., ‘Functions with strongly closed graphs’, Boll. Un. Mat. Ital 12 (1975), 381–384.Google Scholar
[35]Long, Paul E. and Herrington, Larry L., ‘Para-continuous functions’, Egypt, Proc. Math. Phys. Soc. 52 (1985), 1–5.Google Scholar
[36]Malghan, S.R. and Hanchinamani, V.V., ‘N-continuous functions’, Bruxelles, Ann. Soc. Sci. 98 (1984), 69–79.Google Scholar
[37]Mathur, Asha, ‘δ-continuous mappings’, (preprint).Google Scholar
[38]Mrsevic, M. and Reilly, I.L., ‘On N-continuity and co N-closed topologies’, Recerche di Matematica (to appear).Google Scholar
[39]Noiri, Takashi, ‘Between continuity and weak continuity’, Boll. Un. Mat. Ital. 9 (1974), 647–654.Google Scholar
[40]Noiri, Takashi, ‘A remark on almost continuous mappings’, Proc. Japan Acad. 50 (1974), 205–207.Google Scholar
[41]Noiri, Takashi, ‘On functions with strongly closed graphs’, Acta Math. Hungar. 32 (1978), 1–4.CrossRefGoogle Scholar
[42]Noiri, Takashi, ‘Properties of H-continuous functions’, Res. Rep. of Yatsushiro National College of Technology 1 (1979), 85–90.Google Scholar
[43]Noiri, Takashi, ‘Properties of almost c-continuous functions’, J. Korean Math. Soc. 15 (1979), 85–90.Google Scholar
[44]Noiri, Takashi, ‘δ-continuous functions’, J. Korean Math. Soc. 16 (1980), 161–166.Google Scholar
[45]Noiri, Takashi, ‘Strong forms of continuity in topological spaces’, Rend. Circ. Mat. Palermo (2) Suppl. 12 (1986), 107–113.Google Scholar
[46]Su Park, Yang, ‘c*-continuous functions’, J. Korean Math. Soc. 8 (1971), 69–72.Google Scholar
[47]Porter, J.R. and Thomas, J., ‘On H-closed and minimal Hausdorff spaces’, Trans. Amer. Math. Soc. 18 (1969), 159–170.Google Scholar
[48]Porter, J.R. and Woods, R.G., ‘Ultra-Hausdorff H-closed extensions’, Pacific J. Math. 84 (1979), 399–411.Google Scholar
[49]Rayburn, Marlon C., ‘On hard sets’, Gen. Topol. Appl. 6 (1976), 21–26.Google Scholar
[50]Rose, D.A., ‘On Levine’s decomposition of continuity’, Canad. Math. Bull. 21 (1978), 477–481.Google Scholar
[51]Singal, M.K. and Singal, A.R., ‘Almost continuous functions’, Yokohama Math. J. 16 (1968), 63–73.Google Scholar
[52]Singal, M.K. and Mathur, Asha, ‘Nearly compact spaces’, Boll. Un. Mat. Ital 2 (1969), 702–710.Google Scholar
[53]Singal, M.K. and Niemse, S.B., ‘z-continuous functions’, Yokohama Math. J.Google Scholar
[54]Singal, M.K. and Jain, R.C., ‘Mildly continuous functions’, (preprint).Google Scholar
[55]Singh, I.J. and Prasad, R., ‘Almost c-continuous functions’, Indian J. Math. 27 (1985), 165–168.Google Scholar
[56]Steen, L.A. and Seebach, J.A. Jr, Counterexamples in topology (Springer-Verlag, Berlin, Heidelberg, New York, 1978).Google Scholar
[57]Stone, M.H., ‘Applications of the theory of Boolean rings to general topology’, Trans. Amer. Math. Soc. 41 (1937), 375–481.CrossRefGoogle Scholar
[58]Suk, G.H., ‘Almost c-continuous functions’, J. Korean Math. Soc. 14 (1978), 229–234.Google Scholar
[59]Tong, J., ‘Weak almost continuous mappings and weak nearly compact spaces’, Bell. Un. Mat. Ital (6) 1-A (1982), 385–381.Google Scholar
[60]Veličko, N.Y., ‘On H-closed spaces’, (Russian), Mat. Sb. 70 (1966), 98–112. translated as, Amer. Math. Soc. Transl. 78 (1968), 103–118.Google Scholar
[61]Wagner, D.W., ‘Semi-compactness with respect to Euclidean cone’, Canad. J. Math. 29 (1977), 29–36.Google Scholar
[62]Whyburn, G.T., ‘Semilocally connected sets’, Amer. J. Math. 61 (1939), 733–741.Google Scholar
[63]Zaičev, V., ‘Some classes of topological spaces and their bicompactifications’, Soviet Math. Dokl. 9 (1968), 192–193.Google Scholar