The continuity axiom and the Čech homology

Closed Coverings in Cech Homology Theory

Transactions of the American Mathematical Society ◽

10.2307/1992817 ◽

1957 ◽

Vol 84 (2) ◽

pp. 319

Author(s):

E. E. Floyd

Keyword(s):

Homology Theory ◽

Čech Homology

The exactness of Čech homology over a vector space

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100035398 ◽

1961 ◽

Vol 57 (2) ◽

pp. 428-429 ◽

Cited By ~ 2

Author(s):

G. M. Kelly

Keyword(s):

Vector Space ◽

Čech Homology

Cosheaves and Čech Homology

Sheaf Theory - Graduate Texts in Mathematics ◽

10.1007/978-1-4612-0647-7_6 ◽

1997 ◽

pp. 417-448

Author(s):

Glen E. Bredon

Keyword(s):

Čech Homology

Cech Homology Characterizations of Infinite Dimensional Manifolds

American Journal of Mathematics ◽

10.2307/2374099 ◽

1981 ◽

Vol 103 (3) ◽

pp. 411 ◽

Cited By ~ 16

Author(s):

Robert J. Daverman ◽

John J. Walsh

Keyword(s):

Infinite Dimensional ◽

Čech Homology

Čech Complexes, Čech Homology and Cohomology

Springer Monographs in Mathematics - Completion, Čech and Local Homology and Cohomology ◽

10.1007/978-3-319-96517-8_6 ◽

2018 ◽

pp. 135-163

Author(s):

Peter Schenzel ◽

Anne-Marie Simon

Keyword(s):

Čech Homology

Generalized functions and Čech homology theory

Annali di Matematica Pura ed Applicata (1923 -) ◽

10.1007/bf02414326 ◽

1966 ◽

Vol 72 (1) ◽

pp. 45-57 ◽

Cited By ~ 4

Author(s):

Rubens G. Lintz

Keyword(s):

Homology Theory ◽

Generalized Functions ◽

Čech Homology

Partial mappings, Čech homology and ordinary differential equations

Topology and its Applications ◽

10.1016/j.topol.2017.02.044 ◽

2017 ◽

Vol 221 ◽

pp. 647-655 ◽

Cited By ~ 2

Author(s):

V.V. Filippov

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Čech Homology

Tarski Geometry Axioms – Part II

Formalized Mathematics ◽

10.1515/forma-2016-0012 ◽

2016 ◽

Vol 24 (2) ◽

pp. 157-166 ◽

Cited By ~ 1

Author(s):

Roland Coghetto ◽

Adam Grabowski

Keyword(s):

Lattice Theory ◽

Euclidean Plane ◽

List Type ◽

Proof Assistant ◽

Lower Dimension ◽

Formal Proofs ◽

Ternary Relation ◽

Alfred Tarski ◽

Congruence Identity ◽

Continuity Axiom

Summary In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness identity (A6), Pasch (A7). Also a simple model, which satisfies these axioms, was previously constructed, and described in [6]. In this paper, we deal with four remaining axioms, namely: the lower dimension axiom (A8), the upper dimension axiom (A9), the Euclid axiom (A10), the continuity axiom (A11). They were introduced in the form of Mizar attributes. Additionally, the relation of congruence of triangles cong is introduced via congruence of sides (SSS). In order to show that the structure which satisfies all eleven Tarski’s axioms really exists, we provided a proof of the registration of a cluster that the Euclidean plane, or rather a natural [5] extension of ordinary metric structure Euclid 2 satisfies all these attributes. Although the tradition of the mechanization of Tarski’s geometry in Mizar is not as long as in Coq [11], first approaches to this topic were done in Mizar in 1990 [16] (even if this article started formal Hilbert axiomatization of geometry, and parallel development was rather unlikely at that time [8]). Connection with another proof assistant should be mentioned – we had some doubts about the proof of the Euclid’s axiom and inspection of the proof taken from Archive of Formal Proofs of Isabelle [10] clarified things a bit. Our development allows for the future faithful mechanization of [13] and opens the possibility of automatically generated Prover9 proofs which was useful in the case of lattice theory [7].

Nonhuman Primates Satisfy Utility Maximization in Compliance with the Continuity Axiom of Expected Utility Theory

Journal of Neuroscience ◽

10.1523/jneurosci.0955-20.2020 ◽

2021 ◽

Vol 41 (13) ◽

pp. 2964-2979 ◽

Cited By ~ 2

Author(s):

Simone Ferrari-Toniolo ◽

Philipe M. Bujold ◽

Fabian Grabenhorst ◽

Raymundo Báez-Mendoza ◽

Wolfram Schultz

Keyword(s):

Expected Utility ◽

Utility Theory ◽

Utility Maximization ◽

Nonhuman Primates ◽

Expected Utility Theory ◽

Continuity Axiom

On Küneth’s correlation and its applications

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2021 ◽

2019 ◽

Vol 26 (2) ◽

pp. 295-301

Author(s):

Leonard Mdzinarishvili

Keyword(s):

Exact Sequence ◽

Hausdorff Space ◽

Chain Complex ◽

Abelian Category ◽

Derived Functor ◽

Direct System ◽

Arbitrary Complex ◽

Formula Group ◽

Čech Homology ◽

A Chain

Abstract Let {\mathcal{K}} be an abelian category that has enough injective objects, let {T\colon\mathcal{K}\to A} be any left exact covariant additive functor to an abelian category A and let {T^{(i)}} be a right derived functor, {u\geq 1} , [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If {T^{(i)}=0} for {i\geq 2} and {T^{(i)}C_{n}=0} for all {n\in\mathbb{Z}} , then there is an exact sequence 0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})% \longrightarrow TH_{n}(C_{*})\longrightarrow 0, where {C_{*}=\{C_{n}\}} is a chain complex in the category {\mathcal{K}} , {H_{n}(C_{*})} is the homology of the chain complex {C_{*}} , {TC_{*}} is a chain complex in the category A, and {H_{n}(TC_{*})} is the homology of the chain complex {TC_{*}} . This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence \displaystyle\cdots\longrightarrow T^{(2i+1)}H_{n+i+1}\longrightarrow\cdots% \longrightarrow T^{(3)}H_{n+2}\longrightarrow T^{(1)}H_{n+1}\longrightarrow H_% {n}(TC_{*}) \displaystyle\longrightarrow TH_{n}(C_{*})\longrightarrow T^{(2)}H_{n+1}% \longrightarrow T^{(4)}H_{n+2}\longrightarrow\cdots\longrightarrow T^{(2i)}H_{% n+i}\longrightarrow\cdots holds, where {T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})} , {T^{(2i)}H_{n+i}=T^{(2i)}H_{n+i}(C_{*})} . The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system {\{U\}} of open subsets U of X such that {\overline{U}} is a compact subset of X.