The Historical Development of Uniform, Proximal, and Nearness Concepts in Topology

For formulating results or definitions from older papers and books we shall mostly use modern terms and notation to avoid misunderstanding.

[1]  T. Shirota A class of topological spaces , 1952 .

[2]  D. Dantzig Zur topologischen Algebra , 1933 .

[3]  T. Dube The Tamano-Dowker type theorems for nearness frames , 1995 .

[4]  C. Goffman,et al.  On the metrization of uniform space , 1950 .

[5]  A. P. Shostak E-compact extensions of topological spaces , 1974 .

[6]  H. Brandenburg On spaces with aGδ-basis , 1980 .

[7]  M. Fréchet,et al.  Les espaces abstraits , 1929 .

[8]  Fletcher Quasi-Uniform Spaces , 1982 .

[9]  R. S. Lee The category of uniform convergence spaces is cartesian closed , 1976, Bulletin of the Australian Mathematical Society.

[10]  S. Fomin,et al.  Extensions of Topological Spaces , 1943 .

[11]  E. E. Reed Nearnesses, proximities, and ₁-compactifications , 1978 .

[12]  Nearness and metrization , 1976 .

[13]  S. A. Naimpally,et al.  Nearness — A Better Approach to Continuity and Limits , 1974 .

[14]  J. W. Carlson Topological properties in nearness spaces , 1978 .

[15]  M. Fréchet Sur quelques points du calcul fonctionnel , 1906 .

[16]  F. Hausdorff Grundzüge der Mengenlehre , 1914 .

[17]  Oswald Wyler Filter space monads, regularity, completions , 1974 .

[18]  A. Frink Distance functions and the metrization problem , 1937 .

[19]  Ákos Császár,et al.  Foundations of general topology , 1963 .

[20]  D. Kent,et al.  Regular completions of Cauchy spaces via function algebras , 1974, Bulletin of the Australian Mathematical Society.

[21]  T. Dube Paracompact and locally fine nearness frames , 1995 .

[22]  H. Herrlich,et al.  Morita-extensions and nearness-completions , 1998 .

[23]  Oswald Wyler,et al.  On the categories of general topology and topological algebra , 1971 .

[24]  H. Hahn Bemerkungen zu den Untersuchungen des Herrn M. Fréchet: Sur quelques points du calcul fonctionnel , 1908 .

[25]  E. Heine Ueber trigonometrische Reihen. , .

[26]  G. Cantor Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen , 1872 .

[27]  A. Davis,et al.  Indexed Systems of Neighborhoods for General Topological Spaces , 1961 .

[28]  Douglas E. Harris Structures in topology , 1971 .

[29]  L. Graves On the Completing of a Hausdorff Space , 1937 .

[30]  H. Bentley HOMOLOGY AND COHOMOLOGY FOR MEROTOPIC AND NEARNESS SPACES , 1983 .

[31]  W. Weiss Representation of merotopic and nearness spaces , 1988 .

[32]  Sur les ensembles abstraits , 1921 .

[33]  A. Császár Proximities, screens, merotopies, uniformities. II , 1987 .

[34]  J. Pelant Combinatorial properties of uniformities , 1977 .

[35]  Gerhard Preub,et al.  SEMIUNIFORM CONVERGENCE SPACES , 1995 .

[36]  N. C. Heldermann COREFLECTIONS INDUCED BY FINE FUNCTORS , 1982 .

[37]  Horst Herrlich,et al.  A concept of nearness , 1974 .

[38]  H. Bentley Nearness Spaces and Extensions of Topological Spaces , 1975 .

[39]  Y. T. Rhineghost Products of quotients in near , 1984 .

[40]  C. E. Aull E.W. Chittenden and the early history of general topology , 1981 .

[41]  John W. Tukey,et al.  Convergence and uniformity in topology , 1940 .

[42]  W. Hunsaker,et al.  Extensions of continuous functions: reflective functors , 1987, Bulletin of the Australian Mathematical Society.

[43]  D. Preiss,et al.  In-between theorems in uniform spaces , 1980 .

[44]  Kiiti Morota Chapter 1 Extensions of Mappings I , 1989 .

[45]  G. Kalisch On uniform spaces and topological algebra , 1946 .

[46]  H. Herrlich,et al.  Improving constructions in topology , 1991 .

[47]  E. W. Chittenden On the equivalence of Écart and voisinage , 1917 .

[48]  G. Preuß Some point set properties of merotopic spaces and their cohomology groups , 1983 .

[49]  Les ensembles abstraits et le calcul fonctionnel , 1910 .

[50]  Sung Sa Hong,et al.  Cauchy completions of nearness frames , 1995, Appl. Categorical Struct..

[51]  P. Samuel,et al.  Ultrafilters and compactification of uniform spaces , 1948 .

[52]  O. Wyler Ein Komplettierungsfunktor für uniforme Limesräume , 1970 .

[53]  H. Herrlich PRODUCTS IN TOPOLOGY , 1977 .

[54]  E. W. Chittenden On the metrization problem and related problems in the theory of abstract sets , 1927 .

[55]  M. Katětov Measures in Fully Normal Spaces , 1951 .

[56]  J. W. Carlson Nearness and quasi-uniform structures , 1983 .

[57]  M. Katětov On Real-Valued Functions in Topological Spaces , 1951 .

[58]  N. C. Heldermann Concentrated nearness spaces , 1979 .

[59]  Susanne Dierolf,et al.  Uniform Structures on Topological Groups and Their Quotients , 1982 .

[60]  M. Hastings On heminearness spaces , 1982 .

[61]  Eva Lowen-Colebunders Function Classes of Cauchy Continuous Maps , 1989 .

[62]  H. Freudenthal,et al.  Neuaufbau Der Endentheorie , 1942 .

[63]  J. W. Carlson H-closed and countably compact extensions , 1979 .

[64]  F. Schwarz CARTESIAN CLOSEDNESS, EXPONENTIALITY, AND FINAL HULLS IN PSEUDOTOPOLOGICAL SPACES , 1982 .

[65]  N. Aronszajn,et al.  EXTENSION OF UNIFORMLY CONTINUOUS TRANSFORMATIONS AND HYPERCONVEX METRIC SPACES , 1956 .

[66]  B. Banaschewski,et al.  CAUCHY POINTS OF UNIFORM AND NEARNESS FRAMES , 1996 .

[67]  A. Hager Some Nearly Fine Uniform Spaces , 1974 .

[68]  P. Dugac Eléments d'analyse de Karl Weierstrass , 1973 .

[69]  C. H. Dowker Mappings of proximity structures , 1962 .

[70]  C. H. Dowker HOMOLOGY GROUPS OF RELATIONS , 1952 .

[71]  Takeshi Inagaki SUR LES ESPACES À STRUCTURE UNIFORME , 1943 .

[72]  R. Lowen Approach Spaces A Common Supercategory of TOP and MET , 1989 .

[73]  Ellen E. Reed,et al.  Completions of uniform convergence spaces , 1971 .

[74]  M. D. Rice,et al.  Productivity of coreflective subcategories of uniform spaces , 1978 .

[75]  Jan Pelant,et al.  Remark on locally fine spaces , 1975 .

[76]  Ludwig Scheeffer,et al.  Zur Theorie der Stetigen Funktionen einer Reellen Veränderlichen , 1884 .

[77]  HAUSDORFF NEARNESS SPACES , 1979 .

[78]  H. L. Smith,et al.  A General Theory of Limits , 1922 .

[79]  J. W. Carlson B-completeness in nearness spaces , 1975 .

[80]  G. Vidossich Two remarks on A. Gleason's factorization theorem , 1970 .

[81]  C. H. Cook,et al.  Uniform convergence structures , 1967 .

[82]  Closed structures on categories of topological spaces , 1985 .

[83]  Gary Richardson,et al.  Cauchy spaces with regular completions , 1984 .

[84]  Miroslav Katětov,et al.  On continuity structures and spaces of mappings , 1965 .

[85]  Petr Simon,et al.  On local merotopic character , 1971 .

[86]  K. Chattopadhyay,et al.  Completion of merotopic spaces and extension of uniformly continuous maps , 1983 .

[87]  W. Gähler,et al.  Completion of Pseudo‐Topological Vector Spaces , 1976 .

[88]  H. Bentley The role of nearness spaces in topology , 1976 .

[89]  Guillaume C L Brümmer A categorial study of initiality in uniform topology , 1971 .

[90]  M. Katětov Entropies of self-mappings of topological spaces with richer structures , 1993 .

[91]  H. Bentley NORMAL NEARNESS SPACES , 1977 .

[92]  H. Herrlich Some topological theorems which fail to be true , 1976 .

[93]  S. A. Naimpally,et al.  Reflective functors via nearness , 1974 .

[94]  H. Herrlich Are there convenient subcategories of Top , 1983 .

[95]  H. Brandenburg On para-uniform nearness spaces andD-complete regularity , 1988 .

[96]  Garrett Birkhoff,et al.  A note on topological groups , 1936 .

[97]  Horst Herrlich,et al.  Topologische Reflexionen und Coreflexionen , 1968 .

[98]  W. J. Thron,et al.  Merotopic Spaces and Extensions of Closure Spaces , 1983, Canadian Journal of Mathematics.

[99]  Bernard Bolzano Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reele Wurzel der Gleichung liege , .

[100]  H. Herrlich,et al.  The reals and the reals , 1978 .

[101]  H. H. Keller,et al.  Die Limes-Uniformisierbarkeit der Limesräume , 1968 .

[102]  J. Fenstad Onl-groups of uniformly continuous functions. I approximation theory , 1963 .

[103]  John von Neumann,et al.  On complete topological spaces , 1935 .

[104]  I. Konishi On uniform topologies in general spaces , 1952 .

[105]  D. C. Kent,et al.  Completion functors for Cauchy spaces , 1979 .

[106]  SPACES DEFINED BY A FAMILY OF CENTRED SYSTEMS , 1976 .

[107]  E. Lowen-Colebunders Uniformizable Cauchy spaces , 1982 .

[108]  Sur les ensembles abstraits (extrait d'une lettre à M. Frechet avril 1922) , 1924 .

[109]  R. Doss On Uniform Spaces with a Unique Structure , 1949 .

[110]  Gerhard Preuss,et al.  Theory of topological structures , 1987 .

[111]  Maryam Hastings Open Problem A , 1989 .

[112]  M. Husek Products of uniform spaces , 1979 .

[113]  On the space of subsets of a uniform space , 1961 .

[114]  J. Dieudonné,et al.  Sur les espaces uniformes complets , 1939 .

[115]  P. S. Aleksandrov,et al.  THE MAIN ASPECTS IN THE DEVELOPMENT OF SET-THEORETICAL TOPOLOGY , 1978 .

[116]  Horst Herrlich,et al.  Some open categorical problems inTop , 1993, Appl. Categorical Struct..

[117]  Seymour Ginsburg,et al.  SOME OPERATORS ON UNIFORM SPACES , 1959 .

[118]  G. Birkhoff Moore-Smith Convergence in General Topology , 1937 .

[119]  M. Sugiura Correction to my paper: Conjugate classes of Cartan subalgebras in real semisimple Lie algebras , 1971 .

[120]  A. Weil,et al.  Sur les espaces à structure uniforme et sur la topologie générale , 1939 .

[121]  J. W. Carlson PRIME WALLMAN COMPACTIFICATION , 1987 .

[122]  Ulisse Dini,et al.  Fondamenti per la teorica delle funzioni di variabili reali , 1878 .

[123]  F. Husdorff Die Mengen $G_δ$ in vollständigen Räumen , .

[124]  S. Naimpally,et al.  Not every near family is contained in a clan , 1975 .

[125]  D. Kent,et al.  Regular completions of Cauchy spaces , 1974 .

[126]  H. Herrlich,et al.  Convenient categories for topologists , 1976 .

[127]  A. H. Stone UNIVERSAL SPACES FOR SOME METRIZABLE UNIFORMITIES , 1960 .

[128]  W. J. Thron,et al.  Nearness structures and proximity extensions , 1975 .

[129]  Shizuo Kakutani Über die Metrisation der topologischen Gruppen , 1936 .

[130]  H. Herrlich,et al.  Categorical Topology — Its Origins, as Exemplified by the Unfolding of the Theory of Topological Reflections and Coreflections before 1971 , 1997 .

[131]  J. Pelant Locally fine uniformities and normal covers , 1987 .

[132]  Leonard Gillman,et al.  Rings of continuous functions , 1961 .