Pointless topology

# Pointless topology

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, pointless topology (also called point-free or pointfree topology) is an approach to topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

that avoids mentioning points. The name 'pointless topology' is due to John von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...

. The ideas of pointless topology are closely related to mereotopologies
Mereotopology
In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts....

in which regions (sets) are treated as foundational without explicit reference to underlying point sets.

## General concepts

Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

consists of a set of points, together with a system of open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

s. These open sets with the operations of intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

and union
Union (set theory)
In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

form a lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

of pointless topological spaces, also called locales, as an extension of the category of ordinary topological spaces.

## Categories of frames and locales

Formally, a frame is defined to be a lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

L in which finite meets distribute
Distributivity (order theory)
In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima...

over arbitrary joins, i.e. every (even infinite) subset {ai} of L has a supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

⋁ai such that

for all b in L. These frames, together with lattice homomorphisms that respect arbitrary suprema, form a category. The dual
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...

of the category of frames is called the category of locales and generalizes the category Top
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...

of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every continuous map between topological spaces X and Y induces a map between the lattices of open sets in the opposite direction as for every continuous function fX → Y and every open set O in Y the inverse image f -1(O) is an open set in X.

## Relation to point-set topology

It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. While many important theorems in point-set topology require the axiom of choice, this is not true for some of their analogues in locale theory. This can be useful if one works in a topos
Topos
In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...

that does not have the axiom of choice.

The concept of "product of locales" diverges slightly from the concept of "product of topological spaces
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...

", and this divergence has been called a disadvantage of the locale approach.
Others claim that the locale product is more natural, and point to several "desirable" properties not shared by products of topological spaces.

For almost all spaces (more precisely for sober space
Sober space
In mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...

s), the topological product and the localic product have the same set of points. The products differ in how equality between sets of open rectangles, the canonical base for the product topology, is defined: equality for the topological product means the same set of points is covered;
equality for the localic product means provable equality using the frame axioms. As a result, two open sublocales of a localic product may contain exactly the same points without being equal.

A point where locale theory and topology diverge much more strongly is the concept of subspaces vs. sublocales.
The rational numbers have c subspaces but 2c sublocales. The proof for the latter statement is due to John Isbell, and uses the fact that the rational numbers have c many pairwise almost disjoint (= finite intersection) closed subspaces.

• Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...

. A locale is a complete Heyting algebra
Complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra which is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames...

.
• Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between sober space
Sober space
In mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...

s and spatial locales, are to be found in the article on Stone duality
Stone duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation...

.
• Point-free geometry
• Mereology
Mereology
In philosophy and mathematical logic, mereology treats parts and the wholes they form...

• Mereotopology
Mereotopology
In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts....

• Tacit programming
Tacit programming
Tacit programming is a programming paradigm in which a function definition does not include information regarding its arguments, using combinators and function composition instead of variables...