zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Isbell-hull of a di-space. (English) Zbl 1245.54023

Recall that a pair (X,d) is called a T 0 -quasi-metric space if X is a set and d:X×X[0,) is a mapping satisfying

(i) d(x,x)=0 for each xX;

(ii) d(x,z)d(x,y)+d(y,z) for any x,y,zX; and

(iii) d(x,y)=0=d(y,x) implies that x=y.

A T 0 -quasi-metric space M is said to be injective if it has the property that whenever X is a T 0 -quasi-metric space, A is a subspace of X, and f:AM is a nonexpansive map, then f can be extended to a nonexpansive map g:XM. It is shown that, for every T 0 -quasi-metric space X, there exists a T 0 -quasimetric space I(X) with the following properties:

(a) I(X) is injective:

(b) X is isometric to a subspace of I(X); and

(c) I(X) is minimal with respect to (a) and (b), i.e., whenever M is an injective T 0 -quasi-metric space containing an isometric copy of X, then M contains an isometric copy of I(X).

The construction of I(X) is similar to J. R. Isbell’s construction of the injective hull of a metric space X in [Comment. Alath. Helv. 39, 65–76 (1964; Zbl 0151.30205)] (which is also known as the tight span of X or as the hyperconvex hull of X).

54D35Extensions of topological spaces (compactifications, supercompactifications, completions, etc.)
54E35Metric spaces, metrizability
54E50Complete metric spaces
54C15Retractions of topological spaces
[1]Abreu, T.; Corbacho, E.; Tarieladze, V.: Uniform type hyperspaces, Math. pannon. 19, 155-170 (2008) · Zbl 1199.22003
[2]Aronszajn, N.; Panitchpakdi, P.: Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6, 405-439 (1956) · Zbl 0074.17802
[3]Deza, M. M.; Deza, E.: Encyclopedia of distances, (2009)
[4]Dress, A. W. M.: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. math. 53, 321-402 (1984) · Zbl 0562.54041 · doi:10.1016/0001-8708(84)90029-X
[5]Espínola, R.; Khamsi, M. A.: Introduction to hyperconvex spaces, , 391-435 (2001) · Zbl 1029.47002
[6]Fletcher, P.; Lindgren, W. F.: Quasi-uniform spaces, (1982)
[7]Isbell, J. R.: Six theorems about injective metric spaces, Comment. math. Helv. 39, 65-76 (1964) · Zbl 0151.30205 · doi:10.1007/BF02566944
[8]Jawhari, E. M.; Pouzet, M.; Misane, D.: Retracts: graphs and ordered sets from the metric point of view, Contemp. math. 57, 175-226 (1986) · Zbl 0597.54028
[9]Khamsi, M. A.; Kirk, W. A.: An introduction to metric spaces and fixed point theory, (2001)
[10]Künzi, H. -P.A.; Kivuvu, C. Makitu: A double completion for an arbitrary T0-quasi-metric space, J. log. Algebr. program. 76, 251-269 (2008) · Zbl 1211.54039 · doi:10.1016/j.jlap.2008.02.006
[11]Künzi, H. -P.A.: An introduction to quasi-uniform spaces, Contemp. math. 486, 239-304 (2009) · Zbl 1193.54014
[12]Lowen, R.; Sioen, M.: A unified functional look at completion in MET, UNIF and AP, Appl. categ. Structures 8, 447-461 (2000) · Zbl 0987.54021 · doi:10.1023/A:1008710500986
[13]Richman, F.: The fundamental theorem of algebra: a constructive development without choice, Pacific J. Math. 196, 213-230 (2000) · Zbl 1046.03036 · doi:10.2140/pjm.2000.196.213
[14]Rodríguez-López, J.; Romaguera, S.: Hypertopologies and asymmetric topology, Quad. mat. 22, 317-364 (2009) · Zbl 1221.54014
[15]Rodríguez-López, J.; Sánchez-Granero, M. A.: Some properties of bornological convergences, Topology appl. 158, 101-117 (2011) · Zbl 1213.54035 · doi:10.1016/j.topol.2010.10.009
[16]Rodríguez-López, J.; Schellekens, M. P.; Valero, O.: An extension of the dual complexity space and an application to computer science, Topology appl. 156, 3052-3061 (2009) · Zbl 1198.68153 · doi:10.1016/j.topol.2009.02.009
[17]Romaguera, S.; Valero, O.: Domain theoretic characterisations of quasi-metric completeness in terms of formal balls, Math. structures comput. Sci. 20, 453-472 (2010) · Zbl 1193.54016 · doi:10.1017/S0960129510000010
[18]Salbany, S.: Injective objects and morphisms, , 394-409 (1989)