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Applications of cohomology to questions in set theory i: hausdorff gaps [article]

Daniel Talayco
1993 arXiv   pre-print
The cohomology theory is introduced with enough generality to be applicable to other questions in set theory.  ...  We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps.  ...  In a future paper, we will examine the case of ω 1 -trees and develop a cohomology theory for a class of Aronszajn trees.  ... 
arXiv:math/9311205v1 fatcat:tmwtogf3rzapdmc2w3ol3krfsy

Master index of volumes 51–60

1994 Topology and its Applications  
ELSEVIER Topology and its Applications 60 (1994) 297-304 Elsevier Science B.V.  ...  ., Completions of ranked spaces and their applications to the study of nonabsolute integrals Nail, V.C., Partial confluence and closed subsets of a graph Nel, L.D., see Bonenfant, P.  ...  C.L., Determining topologies by collections of proper maps and by weak topologies Cortiiias, G., L-theory and dihedral homology II Katz, Equivariant semicharacteristics and induction Dqbski,  ... 
doi:10.1016/0166-8641(94)90010-8 fatcat:qluxdsirkbguvow77rovlp6bp4

Page 904 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
branches of the tree corresponding to the con- dition p’.  ...  The first derived limit lim”) of of this inverse system is an object of interest in several areas of mathematics and has its origin in cohomology theory.  ... 

The Cohomology of the Ordinals I: Basic Theory and Consistency Results [article]

Jeffrey Bergfalk, Chris Lambie-Hanson
2019 arXiv   pre-print
This discussion occupies the first half of our paper and is written with a general mathematical audience in mind. We turn in the paper's second half to more properly set-theoretic considerations.  ...  We show in particular that the Čech cohomology groups of the ordinals articulate higher-dimensional generalizations of Todorcevic's walks and coherent sequences techniques, and begin to account for those  ...  Portions of this material appeared in the first author's Ph.D. thesis; he would like to thank the members of his committee -Justin Moore, Slawomir Solecki, and Jim West -as well as Stevo Todorcevic, for  ... 
arXiv:1902.02736v2 fatcat:anqluhzlpjbazk35vxvdq3leyy

The first omega alephs: from simplices to trees of trees to higher walks [article]

Jeffrey Bergfalk
2021 arXiv   pre-print
The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of ℵ_n is n+1.  ...  sets, nontrivial n-coherent families of functions, and higher-dimensional generalizations of portions of Todorcevic's walks technique.  ...  The periodic stimulus of conversations with Stevo Todorcevic has also richly informed this work.  ... 
arXiv:2008.03386v2 fatcat:z352uqtqi5f2hnllweqz4obege

Open problems in topology

Elliott Pearl
2004 Topology and its Applications  
In the notes to this problem, Dow conjectured that there is a model satisfying that if X is compact and ω × X has remote points then X has an open subset with countable cellularity.  ...  Kunen [15] proved that if κ is a regular cardinal, then there is a weak P κ + -point in U(κ), the space of uniform ultrafilters on κ. Problem 5 only asked for the case κ = ω 1 .  ...  Also, Tree gave a ZFC example of a nonmetrizable pseudonormal Moore manifold (which is listed as a subproblem to Problem 314). Reed would like to note that P.  ... 
doi:10.1016/s0166-8641(03)00183-4 fatcat:hx5owowk25ftxducz667cxupiu

Problems from Topology Proceedings [article]

Elliott Pearl
2003 arXiv   pre-print
Arhangelskii's Structure and classification of topological spaces and cardinal invariants (1978); Continuum theory problems by Wayne Lewis (1983); Problems in continuum theory by Janusz R.  ...  's that have some connection to the journal.  ...  These ideas may have application again in continua theory. Here is another application to the zero-dimensional case.  ... 
arXiv:math/0312456v1 fatcat:g5aorxz3z5btlbxoawere25mdy

Complete normality and metrization theory of manifolds

Peter J. Nyikos
2002 Topology and its Applications  
A space is collectionwise Hausdorff (cwH) if every closed discrete subspace D can be expanded to a disjoint collection of open sets each of which meets D in one point.  ...  A manifold is a connected Hausdorff space in which every point has a neighborhood homeomorphic to Euclidean n-space (n is unique).  ...  Then, in 1978, Rudin showed that the existence of perfectly normal nonmetrizable manifolds was independent of the usual axioms of set theory [6] , by showing that they do not exist under MA(ω 1 ).  ... 
doi:10.1016/s0166-8641(01)00181-x fatcat:76ku26s6grg2hgzpw6hpub6pwq

Structurable equivalence relations

Ruiyuan Chen, Alexander S. Kechris
2018 Fundamenta Mathematicae  
For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class.  ...  Finally, we consider the effect on K-structurability of various model-theoretic properties of K.  ...  Kechris-Solecki-Todorcevic [KST, 7 .1] proved a universality result for theories of graphs, which was then extended to arbitrary theories by Miller; see Corollary 4.4.  ... 
doi:10.4064/fm428-7-2017 fatcat:3wv3gcyaq5gupp7v5yw3464emq

Structurable equivalence relations [article]

Ruiyuan Chen, Alexander S. Kechris
2018 arXiv   pre-print
For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class.  ...  Finally, we consider the effect on K-structurability of various model-theoretic properties of K.  ...  We are also grateful to Anush Tserunyan for extensive comments and suggestions, including spotting and correcting an error in the original version of Lemma 8.3.  ... 
arXiv:1606.01995v5 fatcat:b5qy5eim4fgp5jjcplmepwgrgi

Dimensions of ordinals: set theory, homology theory, and the first omega alephs

Jeffrey Bergfalk
2018
Some of their contributions will grow plainer in the pages below. The organizing and overarching intellectual debt is very obviously to the works of Stevo Todorcevic and Sibe Mardešić.  ...  Fundamental to all my thinking, finally, are three teachers each of whom is of a sort that one finds, with luck, maybe once in a lifetime.  ...  The standing kindnesses of parents come in many ways first, and condition all the rest.  ... 
doi:10.7298/x4w37tkk fatcat:r2bivg24rfgr3dlnfiwygqsvaq

Sites whose topoi are the smooth representations of locally profinite groups [article]

Satoshi Kondo, Seidai Yasuda
2017 arXiv   pre-print
We define a class of sites such that the associated topos is equivalent to the category of smooth sets (representations) of some locally prodiscrete monoids (to be defined).  ...  We then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group.  ...  We took the term 'smooth' from the representation theory of locally profinite groups, which is known for its applications in number theory (see Remark 5.7.8) .  ... 
arXiv:1506.08023v2 fatcat:c2o4krhudbcjffsraxc3eauyrm