In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions.[1] In this approach it becomes possible to construct topologically interesting spaces from purely algebraic data.[2]
^Johnstone 1983, p. 41.
^Johnstone 1983, p. 42.
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mathematics, pointlesstopology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning...
general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It...
Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointlesstopology one considers instead the lattice of open...
be paracompact. The notion of paracompact space is also studied in pointlesstopology, where it is more well-behaved. For example, the product of any number...
Complete Heyting algebras thus become a central object of study in pointlesstopology. Every Heyting algebra whose set of non-greatest elements has a greatest...
Marshall Stone. Stone-type dualities also provide the foundation for pointlesstopology and are exploited in theoretical computer science for the study of...
form the foundation of pointlesstopology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms...
entirely constructively. It also produced a more accessible spin-off in pointlesstopology, where the locale concept isolates some insights found by treating...
memory access Locale (mathematics), a complete Heyting algebra used in pointlesstopology Locale (geographic), a geographic place where there is or was human...
Descent (category theory) Grothendieck topology Introduction to topos theory Subobject classifier Pointlesstopology Heyting algebra History of category...
sheaf theory, with geometric origins, and leads to ideas such as pointlesstopology. Categorical logic is now a well-defined field based on type theory...
independence of choice from ZF, as well as providing the framework for pointlesstopology and Stone spaces. An active area of research is the univalent foundations...
topology. Beyond these relations, topology can be looked at solely in terms of the open set lattices, which leads to the study of pointlesstopology....
subsets of X to determine X up to homeomorphism, which is relevant to pointlesstopology. Sobriety makes the specialization preorder a directed complete partial...
Weibel; History of homological algebra Peter Johnstone; The point of pointlesstopology Stasheff, Jim (January 21, 1996). "The Pre-History Of Operads". In...
between finite partial orders and finite distributive lattices. In pointlesstopology the category of spatial locales is known to be equivalent to the dual...
discussed in Casati and Varzi (1999: ch. 5). Mereology Pointlesstopology Point-set topologyTopology Topological space (with links to T0 through T6) Whitehead's...
mathematician working in topology and functional analysis. Her doctoral thesis was one of the initial sources of pointlesstopology. She has also been active...
algebra. Locales are also called frames and appear in Stone duality and pointlesstopology. Locally finite poset. A partially ordered set P is locally finite...
famous duality of sober spaces and spatial locales, exploited in pointlesstopology. Every partially ordered set can be viewed as a category (where the...
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