×

Asymptotic resemblance. (English) Zbl 1355.53019

Proximity relations as studied by Efremovic allow one to model the notion of nearness for subsets of a set, \(X\). Often \(X\) is a metric space, and the relation is thought of as revealing the ‘small scale’ structure of \(X\). This has a variety of uses.
To study the ‘large scale’ structure of spaces, for instance in coarse geometry, one often uses the notion of a coarse structure on \(X\), as defined by Roe.
The aim of this paper is to introduce and study a large scale counterpart of proximity, which is given the name of ‘asymptotic resemblance’. If \(X\) is a set, a binary relation \(\lambda\) on the powerset of \(X\) is called an asymptotic resemblance if it is an equivalence relation on the powerset of \(X\) and, in addition, it satisfies two conditions: (i) if \(A_1\lambda B_1\) and \(A_2\lambda B_2\) then \((A_1\cup A_2)\lambda (B_1\cup B_2)\), and (ii) if \((B_1\cup B_2)\lambda A\) and \(B_1\), \(B_2\) are non empty, then there are non-empty subsets \(A_1\) and \(A_2\) such that \(A=A_1\cup A_2\) and \(B_i\lambda A_i\) for \(i=1,2\).
The properties of this resemblance relation are explored in some depth. Several generic families of examples are given and the relationship with coarse structures is examined.

MSC:

53C05 Connections (general theory)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
51F99 Metric geometry
54C20 Extension of maps
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] G. Bell and A. Dranishnikov, Asymptotic dimension , Topol. Appl. 155 (2008), 1265-1296. · Zbl 1149.54017
[2] A. Dranishnikov, On asymptotic inductive dimension , J. Geom. Topol. 1 (2001), 239-247. · Zbl 1059.54024
[3] J. Dydak and C.S. Hoffland, An alternative definition of coarse structures , Topol. Appl. 155 (2008), 1013-1021. · Zbl 1145.54032
[4] V.A. Efremovic, Infinitesimal space , Dokl. Akad. Nauk. 76 (1951), 341-343.
[5] V.A. Efremovic, The geometry of proximity , I, Mat. Sbor. 31 (1951), 189-200.
[6] S.A. Naimpally and B.D. Warrack, Proximity spaces , Cambr. Tract Math. 59 , Cambridge University Press, Cambridge, UK, 1970. · Zbl 0206.24601
[7] I. Protasov, Normal ball structures , Mat. Stud. 20 (2003), 3-16. · Zbl 1053.54503
[8] J. Roe, Lectures on coarse geometry , Univ. Lect. Series 31 , American Mathematical Society, Providence, RI, 2003. · Zbl 1042.53027
[9] J.W. Tukey, Convergence and uniformity in topology , Ann. Math. Stud. 2 , Princeton University Press, Princeton, 1940. · Zbl 0025.09102
[10] A. Weil, Sur les espaces a structure uniforme et sur la topologie generale , Herman, Paris, 1937. · JFM 63.0569.04
[11] S. Willard, General topology , Addison-Wesley, Reading, MA, 1970.
[12] N. Wright, Simultaneous metrizability of coarse spaces , Proc. Amer. Math. Soc. 139 (2011), 3271-3278. · Zbl 1232.46065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.