Dyadic space

Last updated

In mathematics, a dyadic compactum is a Hausdorff topological space that is the continuous image of a product of discrete two-point spaces, [1] and a dyadic space is a topological space with a compactification which is a dyadic compactum. [2] However, many authors use the term dyadic space with the same meaning as dyadic compactum above. [3] [4] [5]

Dyadic compacta and spaces satisfy the Suslin condition, and were introduced by Russian mathematician Pavel Alexandrov. [1] Polyadic spaces are generalisation of dyadic spaces. [5]

Related Research Articles

In topology and related branches of mathematics, a Hausdorff space ( HOWSS-dorf, HOWZ-dorf), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.

<span class="mw-page-title-main">Algebraic topology</span> Branch of mathematics

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.

In mathematics, a topological ring is a ring that is also a topological space such that both the addition and the multiplication are continuous as maps:

In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour.

In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras.

<span class="mw-page-title-main">Discrete group</span>

In mathematics, a topological group G is called a discrete group if there is no limit point in it. Equivalently, the group G is discrete if and only if its identity is isolated.

In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism

In functional analysis and related areas of mathematics, a Montel space, named after Paul Montel, is any topological vector space (TVS) in which an analog of Montel's theorem holds. Specifically, a Montel space is a barrelled topological vector space in which every closed and bounded subset is compact.

In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.

In mathematics, a Cantor cube is a topological group of the form {0, 1}A for some index set A. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2.

In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of pseudocompactness. Pseudocompact spaces were defined by Edwin Hewitt in 1948.

In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.

In algebraic topology, a Poincaré space is an n-dimensional topological space with a distinguished element µ of its nth homology group such that taking the cap product with an element of the kth cohomology group yields an isomorphism to the (n − k)th homology group. The space is essentially one for which Poincaré duality is valid; more precisely, one whose singular chain complex forms a Poincaré complex with respect to the distinguished element µ.

In mathematics an Eberlein compactum, studied by William Frederick Eberlein, is a compact topological space homeomorphic to a subset of a Banach space with the weak topology. Every compact metric space, more generally every one-point compactification of a locally compact metric space, is Eberlein compact. The converse is not true.

Alexander Vladimirovich Arhangelskii is a Russian mathematician. His research, comprising over 200 published papers, covers various subfields of general topology. He has done particularly important work in metrizability theory and generalized metric spaces, cardinal functions, topological function spaces and other topological groups, and special classes of topological maps. After a long and distinguished career at Moscow State University, he moved to the United States in the 1990s. In 1993 he joined the faculty of Ohio University, from which he retired in 2011.

In the mathematical field of topology, there are various notions of a P-space and of a p-space.

<span class="mw-page-title-main">Isaac Namioka</span> Japanese-American mathematician (1928–2019)

Isaac Namioka was a Japanese-American mathematician who worked in general topology and functional analysis. He was a professor emeritus of mathematics at the University of Washington. He died at home in Seattle on September 25, 2019.

In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.

In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation is orbit-equivalent to a Markov odometer.

References

  1. 1 2 Efimov, B.A. (2001) [1994], "Dyadic compactum", Encyclopedia of Mathematics , EMS Press
  2. Efimov, B.A. (2001) [1994], "Dyadic space", Encyclopedia of Mathematics , EMS Press
  3. Engelking, Ryszard (1977). General Topology. Monografie Matematyczne. Vol. 60. Warsaw: PWN. p. 231. Zbl   0373.54002.
  4. T. C. Przymusinski, Products of normal spaces, Ch. XVIII In K. Kunen and J.E. Vaughan (eds) Handbook of Set-Theoretic Topology. North-Holland, Amsterdam, 1984, p. 794.
  5. 1 2 Hart, Klaas Pieter; Nagata, Jun-iti; Vaughan, Jerry E. (2003). Encyclopedia of General Topology. Elsevier Science. pp. 13, 193. ISBN   978-0444503558.