In mathematics, a generalized space is a generalization of 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 points ...
. Impetuses for such a generalization comes at least in two forms:
# A desire to apply concepts like
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
for objects that are not traditionally viewed as spaces. For example, a
topos was originally introduced for this reason.
# A practical need to remedy the deficiencies that some naturally-occurring categories of spaces (e.g., ones in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
) tend not to be
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
, a standard requirement to do homological algebra.
Alexander Grothendieck's dictum says a topos is a generalized space; precisely, he and his followers write in exposé 4 of SGA I:
However,
William Lawvere argues in his 1975 paper that this dictum should be turned backward; namely, "a topos is the 'algebra of continuous (set-valued) functions' on a generalized space, not the generalized space itself."
A generalized space should not be confused with a geometric object that can substitute the role of spaces. For example, a
stack
Stack may refer to:
Places
* Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group
* Blue Stack Mountains, in Co. Donegal, Ireland
People
* Stack (surname) (including a list of people ...
is typically ''not'' viewed as a space but as a geometric object with a richer structure.
Examples
*A
locale is a sort of a space but perhaps not with enough points. The topos theory is sometimes said to be the theory of generalized locales.
*
Jean Giraud's
gros topos,
Peter Johnstone's topological topos,
or more recent incarnations such as
condensed sets or
pyknotic sets. These attempt to embed the category of (certain) topological spaces into a larger category of generalized spaces, in a way philosophically if not technically similar to the way one generalizes a function to a
generalized function. (Note these constructions are more precise than various
completions of the category of topological spaces.)
References
*
* Lawvere
Categories of spaces may not be generalized spaces as exemplified by directed graphs*
*
{{topology-stub
Mathematical terminology
Topological spaces