Tower extension of topological constructs

De Xue Zhang

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 1, page 41-51
  • ISSN: 0010-2628

Abstract

top
Let L be a completely distributive lattice and C a topological construct; a process is given in this paper to obtain a topological construct 𝐂 ( L ) , called the tower extension of 𝐂 (indexed by L ). This process contains the constructions of probabilistic topological spaces, probabilistic pretopological spaces, probabilistic pseudotopological spaces, limit tower spaces, pretopological approach spaces and pseudotopological approach spaces, etc, as special cases. It is proved that this process has a lot of nice properties, for example, it preserves concrete reflectivity, concrete coreflectivity, and it preserves convenient hulls of topological construct, i.e., the extensional topological hulls (ETH), the cartesian closed topological hulls (CCTH) and the topological universe hulls (TUH) of topological constructs.

How to cite

top

Zhang, De Xue. "Tower extension of topological constructs." Commentationes Mathematicae Universitatis Carolinae 41.1 (2000): 41-51. <http://eudml.org/doc/248610>.

@article{Zhang2000,
abstract = {Let $L$ be a completely distributive lattice and C a topological construct; a process is given in this paper to obtain a topological construct $\mathbf \{C\} (L)$, called the tower extension of $\mathbf \{C\}$ (indexed by $L$). This process contains the constructions of probabilistic topological spaces, probabilistic pretopological spaces, probabilistic pseudotopological spaces, limit tower spaces, pretopological approach spaces and pseudotopological approach spaces, etc, as special cases. It is proved that this process has a lot of nice properties, for example, it preserves concrete reflectivity, concrete coreflectivity, and it preserves convenient hulls of topological construct, i.e., the extensional topological hulls (ETH), the cartesian closed topological hulls (CCTH) and the topological universe hulls (TUH) of topological constructs.},
author = {Zhang, De Xue},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {topological construct; extensionality; cartesian closedness; tower extension; completely distributive lattice; topological category; approach space},
language = {eng},
number = {1},
pages = {41-51},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Tower extension of topological constructs},
url = {http://eudml.org/doc/248610},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Zhang, De Xue
TI - Tower extension of topological constructs
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 1
SP - 41
EP - 51
AB - Let $L$ be a completely distributive lattice and C a topological construct; a process is given in this paper to obtain a topological construct $\mathbf {C} (L)$, called the tower extension of $\mathbf {C}$ (indexed by $L$). This process contains the constructions of probabilistic topological spaces, probabilistic pretopological spaces, probabilistic pseudotopological spaces, limit tower spaces, pretopological approach spaces and pseudotopological approach spaces, etc, as special cases. It is proved that this process has a lot of nice properties, for example, it preserves concrete reflectivity, concrete coreflectivity, and it preserves convenient hulls of topological construct, i.e., the extensional topological hulls (ETH), the cartesian closed topological hulls (CCTH) and the topological universe hulls (TUH) of topological constructs.
LA - eng
KW - topological construct; extensionality; cartesian closedness; tower extension; completely distributive lattice; topological category; approach space
UR - http://eudml.org/doc/248610
ER -

References

top
  1. Adámek J., Herrlich H., Strecker G.E., Abstract and Concrete Categories, Wiley, New York, 1990. MR1051419
  2. Adámek J., Koubek V., Cartesian closed initial completions, Topology Appl. 11 (1980), 1-16. (1980) MR0550868
  3. Antoine P., Étude élémentaire des catégories d'ensembles structrés, Bull. Soc. Math. Belgique 18 (1960), 142-164, 387-414. (1960) 
  4. Blasco N., Lowen R., Fuzzy neighbourhood convergence spaces, Fuzzy Sets and Systems 76 (1995), 395-406. (1995) MR1365406
  5. Bourdaud G., Some cartesian closed topological categories of convergence spaces, in E. Binz, H. Herrlich (eds.), Categorical Topology, (Proc. Mannheim, 1975), Lecture Notes in Mathematics, 540, Springer, Berlin, 1976, pp.93-108. Zbl0332.54004MR0493924
  6. Brock P., Kent D.C., Approach spaces, limit tower spaces and probabilistic convergence spaces, Applied Categorical Structures 5 (1997), 99-110. (1997) Zbl0885.54008MR1456517
  7. Burton M.H., The relationship between a fuzzy uniformity and its family of α -level uniformities, Fuzzy Sets and Systems 54 (1993), 311-315. (1993) Zbl0871.54009MR1215574
  8. Gierz G., et al, A Compendium of Continuous Lattices, Springer, Berlin, 1980. Zbl0452.06001MR0614752
  9. Herrlich H., Cartesian closed topological categories, Math. Colloq. Univ. Cape Town 9 (1974), 1-16. (1974) Zbl0318.18011MR0460414
  10. Herrlich H., Are there convenient subcategories of Top?, Topology Appl. 15 (1983), 263-271. (1983) Zbl0538.18004MR0694546
  11. Herrlich H., Topological improvements of categories of structured sets, Topology Appl. 27 (1987), 145-155. (1987) Zbl0632.54008MR0911688
  12. Herrlich H., Hereditary topological constructs, in Z. Frolík (ed.), General Topology and its relations to Modern Analysis and Algebra VI, Proc. Sixth Prague Topological Symposium, Heldermann Verlag, Berlin, 1988, pp.240-262. Zbl0662.18003MR0952611
  13. Herrlich H., On the representability of partial morphisms in Top and in related constructs, in F. Borceux (ed.), Categorical Algebra and its Applications, (Proc. Louvain-La-Neuve, 1987), Lecture Notes in Mathematics, 1348, Springer, Berlin, 1988, pp.143-153. Zbl0662.18004MR0975967
  14. Herrlich H., Nel L.D., Cartesian closed topological hulls, Proc. Amer. Math. Soc. 62 (1977), 215-222. (1977) Zbl0361.18006MR0476831
  15. Herrlich H., Zhang D., Categorical properties of probabilistic convergence spaces, Applied Categorical Structures 6 (1998), 495-513. (1998) Zbl0917.54003MR1657510
  16. Herrlich H., Lowen-Colebunders E., Schwarz F., Improving Top: PrTop and PsTop, in H. Herrlich, H.E. Porst (eds.), Category Theory at Work, Heldermann Verlag, Berlin, 1991, pp.21-34. Zbl0753.18003MR1147916
  17. Lowen E., Lowen R., Topological quasitopos hulls of categories containing topological and metric objects, Cahiers Top. Géom. Diff. Cat. 30 (1989), 213-228. (1989) Zbl0706.18002MR1029625
  18. Lowen R., Fuzzy uniform spaces, J. Math. Anal. Appl. 82 (1981), 370-385. (1981) Zbl0494.54005MR0629763
  19. Lowen R., Fuzzy neighbourhood spaces, Fuzzy Sets and Systems 7 (1982), 165-189. (1982) 
  20. Lowen R., Approach spaces, a common supercategory of TOP and MET, Math. Nachr. 141 (1989), 183-226. (1989) Zbl0676.54012MR1014427
  21. Lowen R., Approach Spaces, the missing link in the Topology-Uniformity-Metric triad, Oxford Mathematical Monographs, Oxford University Press, 1997. Zbl0891.54001MR1472024
  22. Lowen R., Windels B., AUnif: A common supercategory of pMet and Unif, Internat. J. Math. Math. Sci. 21 (1998), 1-18. (1998) Zbl0890.54024MR1486952
  23. Schwarz F., Description of topological universes, in H. Ehrig et al, (eds.), Categorical Methods in Computor Science with Aspects from Topology, (Proc. Berlin, 1988), Lecture Notes in Computor Science, 393, Springer, Berlin, 1989, pp.325-332. MR1048372
  24. Preuss G., Theory of Topological Structures, an Approach to Categorical Topology, D. Reidel Publishing Company, Dordrecht, 1988. Zbl0649.54001MR0937052
  25. Richardson G.D., Kent D.C., Probabilistic convergence spaces, J. Austral. Math. Soc., (series A) 61 (1996), 400-420. (1996) Zbl0943.54002MR1420347
  26. Wuyts P., On the determination of fuzzy topological spaces, and fuzzy neighbourhood spaces by their level topologies, Fuzzy Sets and Systems 12 (1984), 71-85. (1984) Zbl0574.54004MR0734394
  27. Wyler O., Are there topoi in topology, in E. Binz, H. Herrlich, (eds.), Categorical Topology, (Proc. Mannheim, 1975), Lecture Notes in Mathematics, 540, Springer, Berlin, 1976, pp.699-719. Zbl0354.54001MR0458346
  28. Wyler O., Lecture Notes on Topoi and Quasitopoi, World Scientific, Singapore, 1991. Zbl0727.18001MR1094373

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.