Proximately chain refinable functions

Year 2019, Volume: 48 Issue: 5, 1437 - 1442, 08.10.2019

Abdulla Buklla Gjorgji Markoski

Abstract

We define proximately chain refinable functions as a generalization of refinable maps and investigate some of their properties for Hausdorff paracompact spaces. We prove that the proximate fixed point property is preserved by proximate near homeomorphisms in paracompact Hausdorff spaces. This generalizes a previous result of E. Grace.

Keywords

proximately chain refinable functions, proximate fixed point, near function, chain

References

• [1] D. Buhagiar, T. Miwa and B.A. Pasynkov, Superparacompact type properties, Yoko- hama Math.J. 46, 71–86, 1998.
• [2] R. Engelking, General topology, Heldermann Verlag, 1989.
• [3] J. Ford and J.W. Rogers,Jr., Refinable maps, Colloq.Math. 39, 263–269, 1978.
• [4] E.E. Grace, Refinable maps and the proximate fixed point property, Proceedings of the 1985 topology conference, 293–303, 1985.
• [5] E.E. Grace, Generalized refinable maps, Proc. Amer. Math. Soc. 98, 329–335, 1986.
• [6] E.E. Grace and E.J. Vought, Proximately refinable maps and $\theta_{n}'$-continua, Topology Proc. 15, 39–51, 1990.
• [7] R.W. Kieboom, An intrinsic characterization of the shape of paracompacta by menas of non-continuous single-valued maps, Bull. Belg. Math. Soc. Simon Stevin, 1, 701– 711, 1994.
• [8] V. Klee, Stability of the fixed-point property, Colloq. Math., 8, 43–46, 1961.
• [9] N. Shekutkovski, On the concept of the connectedness, Mat. Bilten, 50 (1), 5–14, 2016.