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Pointless topology
In mathematics, pointless topology (also called point-free or pointfree topology) is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann.〔Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5〕 The ideas of pointless topology are closely related to mereotopologies in which regions (sets) are treated as foundational without explicit reference to underlying point sets.
==General concepts==
Traditionally, a topological space consists of a set of points, together with a system of open sets. These open sets with the operations of intersection and union form a lattice with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the category of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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