Özet
This paper consists of four chapters. The fi rst chapter is an introduction which contains
the basic motivation of nearness theory.
The second section is devoted to nearness in metric spaces. Here, the nearness of two
sets is de fined by gap functional. In particular, the closure point of a set is defi ned using
nearness. The concepts of convergence of a sequence, and continuity of a function are
characterized in terms of nearness. The interior of a set is also defi ned using nearness.
Proximal neighbourhood of a set in a metric space is defi ned and the basic properties
are discussed. Compatible proximity and fine proximity are defi ned and it is proved that
every fine proximity is also metric proximity. For compact spaces, it is shown that every
metric proximity is a fine proximity. The proximal continuity is defi ned and it is proved
that every proximal continuous function is also continuous. The nearness in the sense
of Herrlich is given and the basic properties of Herrlich nearness are presented. Using
metric proximity, a characterization is given for Cauchy sequences. It is proved that
uniform continuity is equivalent to proximal continuity. Further, Hausdorff metric is
de fined and it is proved that every convergent closed set sequence is uniform convergent.
Finally, for continuous extensions, the Taimanov Theorem is proved.
In the third chapter, Efremovic proximity is de fined as a generalization of metric proximity.
Then the Lodato proximity is presented and it is shown that every Efremovic
proximity is a Lodato proximity. Further, compatible proximity is considered in topological
spaces. In this respect, proximity is studied under certain separation properties
of topological spaces. In particular, for completely regular and normal spaces, the existence
of compitable proximity is discussed.
In the fourth chapter, descriptive proximity is discussed in the sense of Efremovic and
Lodato. Finally, spatial and descriptive proximities are compared.
Künye
[1] Anonim, Extension of a Uniformly Continuous Function between Metric
Spaces, http://math.stackexchange.com/questions/245237/extension-of-auniformly-
continuous-function-between-metric-spaces, (Mart, 2017).
[2] B ulb ul, A. Genel Topoloji, Hacettepe Universitesi Yay nlar , G ozden ge cirilmi s
4.bask , Ankara, 2014.
[3] Engelking, R., General Topology, Revised and completed edition, Heldermann Verlag,
Berlin, 1989.
[4] Fr echet, M., Surquelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo,
22, 1-74, 1906.
[5] Gagrat, M., Naimpally, S., Proximity Approach to Semi-Metric and Developable
Spaces. Paci c Journal of Mathematics, 44(1), 93-105, 1973.
[6] Herrlich, H., A concept of nearness. General Topology and Applications, 4, 191-
212, 1974.
[7] Husain, T., Topology and Maps, Plenum Press, New York, 1977.
[8] Naimpally, S., Proximity Approach to Problems in Topology and Analysis, Oldenbourg
Verlag, M unchen, 2009.
[9] Naimpally, S., Peters, J., Topology with Applications: Topological Spaces via Near
and Far, World Scienti c, Singapore, 2013.
[10] Naimpally, S., Warrack, B., Proximity Spaces, Cambridge University Press,
Cambridge, UK, ISBN 978-0-521-09183-1, 1970.
[11] Peters, J., Local Near Sets: Pattern Discovery in Proximity Spaces. Mathematics
in Computer Science, 7, 87-106, 2013.
[12] Peters, J., Near Sets. General theory about nearness of objects, Applied Mathematical
Sciences, 1, no. 53, 2609-2629, 2007.
[13] Peters, J., Topology of Digital Images: Visual Pattern Discovery in Proximity
Spaces, Springer Verlag, Heidelberg, Berlin, 2014.
[14] Peters, J., Naimpally, S., Applications of near sets. Notices of the American Mathematical
Society, 59(4), 536{542, 2012.
[15] Peters, J., Near sets. General theory about nearness of objects. Applied Mathematical
Sciences 1(53), 2609{2629, 2007.
[16] Peters, J., Near sets. Special theory about nearness of objects. Fundam. Inf.
75(1-4), 407{433,2007.
[17] Riesz, F., Stetigkeitbegri und abstrakte mengenlehre. m IV Congresso Internazionale
dei Matematici, 2, 18-24, 1908.
[18] Taimanov, A., On the extension of continuous mapping of topological spaces.
Matematicheskii Sbornik, 31, 451-463, 1952.
