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Dicke, Robert J. (ed.) / Transactions of the Wisconsin Academy of Sciences, Arts and Letters
volume XLIV (1955)

Hammer, Preston C.
General topology, symmetry, and convexity,   pp. 221-255 ff. PDF (10.0 MB)


Page 221

 221GENERAL TOPOLOGY, SYMMETRY, 
AND CONVEXITY 
PRESTON C. HAMMER 
1. Introduction 
 INITIALLY topological concepts were a result of geometrization of analysis
and an analysis of geometry. The studies soon demonstrated the need for more
and more general systems and in this connection the abstract space theory
and abstract general topologies have arisen. The word "abstract"
may be taken
to mean that the elements of the setundeT discussion are not specifically
designatedas long as certain operations are relevant. Thus the points of
an abstract set may be geometrical points, circles, propositions, dishes,
men, or potatoes. 
 "Topology" has been associated with concepts of limit points.,
horneonio.rphisms,
continuity, and related concepts of closed sets, open sets, neighborhoods,
conver gent s equences, connectedness, continua and manifolds. The general
topologies in existence have used special means to introduce the topology
on the basis of axioms r e 1 a t I n g t o closed sets or closure (Kuratowski),
neighborhoods or open sets (Hausdorff, Sierpinski, Alexandroff), directed
sets (E. 
H. Moore, Tukey), and filters (Weil, Bourbaki). The end in view of all such
general definitions seems to beto obtain definitions of limit point and continuity
of transformations and to establish under what conditions special topolo
gi e 8 such as metric topologies may be obtained from those satisfying definitions.
 In contrast to these objectives, we here give direct definitions of closedness,
of closure, and of limit points r e quiring no initial concepts other than
those associated with set algebra and the definition of function. We refuse
to specialize, except in examples, to the topologies which topologists consider
useful since we are here dealing with a topic which belongs in the foundations
of mathematics; which has applications in algebra, geometry, and all mathematics
and logic. We find that the simplest approach is to effectively 


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