constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Domain theory has its origin in
topological algebra concerned with Lawson semilattices,
computer science, concerned with the problem of finding a viable denotational semantics for certain theories of computability (such as the untyped lambda calculus): this resists straightforward interpretations in terms of plain functions between sets but does have interpretation in terms of monotone functions between partially ordered sets (certain lattices) [Scott (1970), Scott & Strachey (1971)].
The theory has since grown into an area which weaves together diverse strands in logic, computability, lattice theory, general topology, and category theory.
Domain theory can be said to have come into existence with the insight from Scott (1970) of interpreting untyped lambda calculus in terms of monotone functions between continuous lattices. In rough terms, the problem can be set out as follows:
Lambda calculus is a syntax of functions and functional application, whose basic constituents are types, terms of types, and type and term formation schemes with which one can speak of product types $A \times B$ and function types $A \Rightarrow B$.
One problem is to construct a meaning or semantics for such syntax in terms of “actual” sets and functions. The categorical semantics of lambda calculus is by cartesian closed categories, in which types are interpreted as objects $A$ and terms are interpreted as (generalized) elements or “functions” $X \to A$.
In so-called untyped lambda calculus (whose syntax is closely connected with the theory of computability and recursive functions), all terms may be regarded as being of the same type $D$ (which therefore need not be mentioned, hence “untyped”), so that, intuitively speaking, elements of $D$ and functions on $D$ are treated on one and the same footing.
The problem Scott solved is to give faithful categorical semantics for untyped lambda calculus; where the basic problem is to construct a cartesian closed category with essentially just one object $D$ (really two objects: $D$ and a terminal object $1$), so that $D$ is closed under formation of products and internal homs: $D \cong D \times D$ and $D \cong D \Rightarrow D$. Notice that in the category of sets, the only solution is to take $D \cong 1$, so that all terms are then equal (“algorithmic inconsistency”). This is not a faithful modeling of untyped lambda calculus, which has provably unequal terms.
In 1969, Dana Scott solved this problem topologically: the terms were interpreted as continuous functions on a suitable space $D$ isomorphic to its own function space. This $D$ is called a domain. Decades later, we now know many techniques for constructing such domains as suitable objects in cartesian closed categories, but Scott’s basic insight, that computability could be interpreted as continuity, continues to exert a decisive influence today.
Martín Escardó writes on the nForum:
The usual practice of domain theory papers and talks is to start by defining “domain” for the purposes of the paper or talk. It is a reusable word. Usually only compound words using the word “domain” have a fixed meaning, such as Scott domain (bounded complete algebraic dcpo?), continuous Scott domain? (bounded complete continuous dcpo), FS domain?, SFP domain?, L-domain?. I wouldn’t use the plain word “domain” in a paper without first explicitly defining it, as it has been used with so many different meanings in the domain theory literature. Also the terminological conventions vary a bit depending on whether domain theory is used for the purposes of computation (e.g., programming language semantics, and theory of computability) or topology and algebra. For example, in programming language semantics papers one often encounters posets that have sups of ascending sequences, rather than directed sets, and a least element, because all the authors are interested in is the existence of (least) fixed point of continuous endo-functions, and these assumptions are enough for this purpose. Such posets are often called domains in such papers. But for applications of domain theory to topology, directed completeness is what one needs in general, and often we have more (even all sups — for example, a topological space is exponentiable if its lattice of open sets is a continuous dcpo — but this dcpo has all sups and moreover is a distributive lattice, and the continuous distributive lattices are precisely the topologies of exponentiable spaces, up to isomorphism). Because these applications and communities are so diverse, it is natural to see a divergence of terminology, even if many of the techniques are the same.
Origin of domain theory in denotational semantics for programming languages:
Dana S. Scott, Outline of a mathematical theory of computation, in: Proceedings of the Fourth Annual Princeton Conference on Information Sciences and Systems (1970) 169–176. [pdf, pdf]
Dana S. Scott, Christopher Strachey, Toward a Mathematical Semantics for Computer Languages, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [pdf, pdf]
Dana Scott, Data types as lattices. SIAM Journal of Computing 5 3 (1976) 522–587 [doi:10.1137/0205037, pdf]
Discussion internal to cartesian closed categories:
Introductions and lecture notes:
Robert D. Tennent, The denotational semantics of programming languages, Communications of the ACM 19 8 (1976) 437–453 [doi:10.1145/360303.360308, pdf]
Andrew M. Pitts, Lecture Notes on Denotational Semantics (2012) [pdf, pdf]
Textbook accounts:
Glynn Winskel, §8 of: The Formal Semantics of Programming Languages, MIT Press (1993) [ISBN:9780262731034, pdf]
David A. Schmidt, Denotational Semantics – A methodology for language development, Allyn and Bacon (1986) [pdf, webpage]
G. Gierz, Karl H. Hofmann, K. Keimel, J. D. Lawson, Michael W. Mislove, Dana S. Scott, Continuous Lattices and Domains, in: Encyclopedia of Mathematics and its Applications 93, Cambridge University Press (2003) [doi:10.1017/CBO9780511542725]
Thomas Streicher, Domain-Theoretic Foundations of Functional Programming, World Scientific (2006) [pdf, doi:10.1142/6284]
See also:
Discussion in homotopy type theory/univalent foundations:
Review:
Samson Abramsky, Achim Jung, Domain Theory, in: Handbook of Logic in Computer Science 3, Oxford University Press (1995) [ISBN:9780198537625, pdf]
Gordon Plotkin, Pisa Notes on Domains [ps]
Possible relation to spacetime causality (cf. causets):
Summary. We discuss the current state of investigations into the domain theoretic structure of spacetime, including recent developments which explain the connection between measurement, the Newtonian concept of time and the Lorentz distance.
Abstract: We prove that a globally hyperbolic spacetime with its causality relation is a bicontinuous poset whose interval topology is the manifold topology. From this one can show that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains. We obtain a mathematical setting in which one can study causality independently of geometry and differentiable structure, and which also suggests that spacetime emerges from something discrete.
In category theoretic generalizations of the notion, the informal idea is to replace partial orders with more general categories, so that there is more space for intensionality. An early reference, which proposes the notion of Scott-complete categories:
The following reference proposes using presheaf categories as a generalization of prime algebraic lattices. This also connects to work on generalized species:
316 1–3 (2004) 153-190 [doi:10.1016/j.tcs.2004.01.029, pdf]
Last revised on January 20, 2024 at 11:09:19. See the history of this page for a list of all contributions to it.