Topologies and smooth maps on initial and final objects in the category of Frolicher spaces

• Autorzy: Batubenge, A. , Tshilombo, H.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we show that when the Frolicher smooth structure is induced on a subset or a quotient set, there are three natural topologies underlying the resulting object. We study these topologies and compare them in each case. It is known that the topology generated by strucure functions is the weakest one in which all functions and curves on the space are continuous. We show that on a subspace, it is rather the trace topology which has this property, while the three topologies are coincident on the quotient space. We construct a base for the Frolicher topology and using either a base or a subbase in the sense of A. Frolicher [9], we characterise the morphisms of this category.

Słowa kluczowe

PL

topologia ogólna; przestrzeń Frolichera

EN

general topology; Frolicher space

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2009
• Tom: Vol. 42, nr 3
• Strony: 641-655
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Batubenge, A.

autor

Tshilombo, H.

• School of Mathematics University of the Witwatersrand Johannesburg Private Bag 3 Wits 2050, South Africa,

Bibliografia

• [1] A. Arkhangel’skii, L. Pontryagin, General Topology I, Springer-Verlag, Berlin 1990.
• [2] A. Batubenge, Symplectic Frölicher Spaces of Constant Dimension, PhD Thesis, University of Cape Town, 2005 (unpublished).
• [3] A. Batubenge, On transversal intersection and gluing symplectic pseudomanifolds, Conference on Symplectic Topology, Stare Jablonki, Poland, 3-10 July 2004 (manusript).
• [4] A. Cap, K-Theory for Convenient Algebras, Thesis, 1993.
• [5] P. Cherenack, Applications of Frölicher spaces to cosmology, Annals Univ. Sc. Budapest 41 (1998), 63-91.
• [6] P. Cherenack, Frölicher versus differential spaces: A prelude to cosmology, G. Brümmer and C. Gilmour (eds), Papers in Honour of Bernhard Banaschewski, 2000 Kluwer Academic Publishers, Printed in the Netherlands (1998), 63-91.
• [7] B. Dugmore, A Framework for Homotopy Theory and Differential Geometry in the Category of Frölicher Spaces, PhD Thesis, University of Cape Town, Cape Town 1999 (unpublished).
• [8] J. Dugundji, Topology, Allyn and Bacon, Inc., Boston 1966.
• [9] A. Frölicher, A. Kriegl, Linear Spaces and Differentiation Theory, Wiley-Interscience, 1971.
• [10] P. Z. Iglesias, Symplectic Diffeology, Private Draft, Typeset, June 2004.
• [11] A. Kriel, P. W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, Volume 53, American Mathematical Society, 1997.
• [12] P. Ntumba, A. Batubenge, On the way to Frölicher Lie groups, Second Edition, Quaestiones Mathematicae 28 (2005), 73-74.
• [13] R. S. Palais, Real Algebraic Differential Topology, Part I, Publish or Perish, Wilmington, 1981.