A Survey on Wallman Bases

Adalberto García-Máynez


Wallman bases are frequently used in compactification processes of topological spaces. However, they are also related with quasi–uniform structures and they are useful to characterize some topological properties. We present a brief survey on the subject which supports these statements.


Wallman basis; Annular basis; Ultrafilter; Perfect extension; Wallman type; Regular Wallman; Equivalent compactifications; Cover uniformity basis; Quasi–uniformity Transitive; Totally bounded; Symmetric; Point symmetric; Locally symmetric

Full Text:



Á. Császár, Fondements de la topologie générale, Budapest-Paris, 1960.

J. Dujundji, Topology, Allyn and Bacon, Inc., Boston, 1966.

A. García–Máynez and A. Tamariz, Topología General, Porrúa, México, 1988.

A. García–Máynez, Property C, Wallman bases and S–metrizability, Topology and its Applications 12 (1981), 237–246. http://dx.doi.org/10.1016/0166-8641(81)90002-X

A. García–Máynez, On Wallman bases and compactifications, Boletín de la Sociedad Matemática Mexicana 3a serie Vol 11, 2 (2005), 283–292.

S. García, Ferreira S. and A. García–Máynez, On weakly-pseudocompact spaces, Houston Journal of Mathematics. 20 (1994), 145–159.

T. Kimura, The Stone– Cech compactifications, the Stone–ˇCech remainder and the regular Wallman property, Proc. Amer. Math. Soc. 99 (1987), 193–198.

H. P. A. Künzi, Topological spaces with a unique compatible quasi–uniformity, Canad. Math. Bull. 29 (1986), 40–43. http://dx.doi.org/10.4153/CMB-1986-007-3

W. J. Pervin, Quasi–uniformization of topological spaces, Math. Ann. 147 (1962), 316–317. http://dx.doi.org/10.1007/BF01440953

S. Romaguera and M. A. Sánchez-Granero, A quasi–uniform Characterization of Wallman type compactifications, Studia Sci. Math. Hungar. 40 (2003), 257–267. http://dx.doi.org/10.1556/SScMath.40.2003.1-2.21

C. R. Solomon, A Hausdorff compactification that is not regular Wallman, General Topology and its Applications 7 (1977), 59–63. http://dx.doi.org/10.1016/0016-660X(77)90007-1

L. A. Steen and J. A. Jr. Seebach, Counterexamples in Topology, Springer-Verlag, New York, 2nd. edition, 1970.

E. F. Steiner, Wallman spaces and compactifications, Fundamenta Mathematicae 61 (1968), 295–304.

A. K. Steiner and E. F. Steiner, Products of compact metric spaces are regular Wallman, Indag. Math. 30 (1968), 428–430.

A. K. Steiner and E. F. Steiner, On the reduction of the Wallman compactification problem to discrete spaces, General Topology and its Applications 7 (1977), 35–37. http://dx.doi.org/10.1016/0016-660X(77)90005-8

V. M. Ul’janov, Solution of the fundamental problem of bicompact extensions of Wallman type, (Russian) Dokl. Akad. Nauk SSSR. 233(1977), 1056–1059. (English translation) Soviet Math. Dokl. 18 (1977), 567–571.

H. Wallman, Lattices and topological spaces, Ann. of Math. 39 (1938), 112–126. http://dx.doi.org/10.2307/1968717

A. Weil, Sur les espaces á structure uniforme et sur la topologie générale, Paris 1938.

Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt