In mathematics, the notion of externology in a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''X'' generalizes the basic properties of the family : ''ε''^''X''_cc = of complements of the

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

subspaces of ''X'', which are used to construct its

Alexandroff compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alex ...

. An externology permits to introduce a notion of end point, to study the divergence of nets in terms of convergence to end points and it is a useful tool for the study and classification of some families of non compact topological spaces. It can also be used to approach a topological space as the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

of other topological spaces: the externologies are very useful when a compact

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

embedded in a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

is approached by its

open neighbourhood In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a ...

Definition

Let (X,τ) be a topological space. An externology on (X,τ) is a non-empty collection ε of open subsets satisfying: * If E₁, E₂ ∈ ε, then E₁ ∩ E₂ ∈ ε; * if E ∈ ε, U ∈ τ and E ⊆ U, then U ∈ ε. An ''exterior space'' (X,τ,ε) consists of a topological space (X,τ) together with an externology ε. An open E which is in ε is said to be an exterior-open subset. A map f:(X,τ,ε) → (X',τ',ε') is said to be an exterior map if it is

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

and f⁻¹(E) ∈ ε, for all E ∈ ε'. The

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

of exterior spaces and exterior maps will be denoted by E. It is remarkable that E is a

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...

and

cocomplete In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in w ...

category.

Some examples of exterior spaces

* For a space (X,τ) one can always consider the ''trivial externology'' ε_tr=, and, on the other hand, the ''total externology'' ε_tot=τ. Note that an externology ε is a

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

if and only if the empty set is a member of ε if and only if ε=τ. * Given a space (X,τ), the externology ε^X_cc of the complements of closed compact subsets of X permits a connection with the theory of

proper map In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. Definition There are several competing def ...

s. * Given a space (X,τ) and a subset A⊆X the family ε(X,A)= is an externology in X. Two particular cases with important applications on shape theory and on

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

, respectively, are the following: * If A is a closed subspace of the

Hilbert cube In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ...

X=Q the externology ε^A=ε(Q,A) is a resolution of A in the sense of the shape theory. * Let X be a

continuous dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...

and P the subset of

periodic point In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given ...

s; we can consider the externology ε(X,P). More generally, if A is an invariant subset the externology ε(X,A) is useful to study the dynamical properties of the

flow Flow may refer to: Science and technology * Fluid flow, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on a set * Flow (psyc ...

Applications of exterior spaces

* ''Proper homotopy theory'': A continuous map f:X→Y between topological spaces is said to be ''

proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...

'' if for every closed compact subset K of Y, f⁻¹(K) is a compact subset of X. The category of spaces and proper maps will be denoted by P. This category and the corresponding proper

homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed b ...

are very useful for the study of non compact spaces. Nevertheless, one has the problem that this category does not have enough limits and colimits and then we can not develop the usual homotopy constructions like loops, homotopy limits and colimits, etc. An answer to this problem is the category of exterior spaces E which admits Quillen model structures and contains as a

full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

the category of spaces and proper maps; that is, there is

full Full may refer to: * People with the surname Full, including: ** Mr. Full (given name unknown), acting Governor of German Cameroon, 1913 to 1914 * A property in the mathematical field of topology; see Full set * A property of functors in the mathe ...

and

faithful Faithful may refer to: Film and television * ''Faithful'' (1910 film), an American comedy short directed by D. W. Griffith * ''Faithful'' (1936 film), a British musical drama directed by Paul L. Stein * ''Faithful'' (1996 film), an American cr ...

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

P→E which carries a topological space (X,τ) to the exterior space (X,τ,ε^X_cc). * ''Proper LS category'': The problem of finding Ganea and Whitehead characterizations of this proper invariant can not be faced within the proper category because of the lack of (co)limits. Nevertheless, an extension of this invariant to the category of exterior spaces permits to find a solution to such a problem. This numerical proper invariant has been applied to the study of open 3-

manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

. * '' Shape theory'': Many

shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

invariants (Borsuk groups, Quigley inward and approaching groups) of a compact

can be obtained as exterior homotopy groups of the exterior space determined by the open neighborhoods of a compact metric space embedded in the Hilbert cube. * ''Discrete and continuous dynamical systems (semi-flows and flows)'': There are many constructions that associate an exterior space to a dynamical system, for example: Given a continuous (discrete) flow one can consider the exterior spaces induced by the open neighborhoods of the subset of

s, Poisson periodic points, omega limits, etc. The constructions and properties of these associated exterior spaces are used to study the dynamical properties of the (semi-flow) flow.

References

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