Giraldo, A. and Rodríguez Sanjurjo, José Manuel (1997) Density and finiteness. A discrete approach to shape. Topology and its Applications, 76 (1). pp. 61-77. ISSN 0166-8641
Restricted to Repository staff only until 31 December 2020.
We show in this paper that the category of shape can be modelled in discrete terms using maps defined in dense subsets of compacta. This approach to shape provides in addition a characterization of the shape of compacta which does not require external elements and uses only continuous single-valued functions, in contrast with the existing internal characterizations of shape. As an application, we prove a connection between the notion of shape image, due to Lisica, and the basic notion of omega limit of a dynamical system.
|Uncontrolled Keywords:||Density: Discrete map: Shape category; Shape image: Omega limit: Lyapunov stability|
|Subjects:||Sciences > Mathematics > Geometry|
Sciences > Mathematics > Topology
N.P. Bhatia and G.P. Szego, Stability theory of dynamical systems, Grundlehren Math. Wiss. 161 (Springer, Berlin, 1970).
K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968) 223-254.
K. Borsuk, On movable compacta, Fund. Math. 66 (1969) 137-146.
K. Borsuk, Theory of Shape. Monogratie Matematyczne 59 (Polish Scientitic Publishers, Warszawa, 1975).
Z. Cerin. Shape theory intrinsically. Publ. Mat. 37 (1993) 317-334.
Z. Cerin. Proximate topology and shape theory, Proc. Roy. Sot. Edinburgh I25 (1995) 595-615.
Z. Cerin. Equivariant shape theory, Math. Proc. Cambridge Philos. Sot. I I7 (1995) 303-320.
J.M. Cordier and T. Porter, Shape Theory. Categorical Methods of Approximation, Ellis Horwood Series: Mathematics and its Applications (Ellis Horwood, Chichester, 1989).
J. Dydak and J. Segal. Shape Theory: An Introduction, Lecture Notes in Math. 688 (Springer, Berlin, 1978).
J. Dydak and J. Segal, A list of open problems in shape theory, in: J. van Mill and G.M. Reed. eds., Open problems in Topology (North-Holland, Amsterdam. 1990) 457-467.
J.E. Felt, E-continuity and shape, Proc. Amer. Math. Sot. 46 (1974) 426430.
R.H. Fox, On shape. Fund. Math. 74 ( 1972) 47-7 I.
A. Giraldo and J.M.R. Sanjurjo, Strong multihomotopy and Steenrod loop spaces, J. Math. Sot. Japan 47 (1995) 475489.
B. Gunther and J. Segal. Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Sot. I I9 (1993) 321-329.
R.W. Kieboom. An intrinsic characterization of the shape of paracompacta by means of noncontinuous single-valued maps. Bull. Belg. Math. Sot. I (1994) 701-71 I.
J.T. Lisica, Strong shape theory and multivalued maps, Glas. Mat. I8 (1983) 371-382.
S. MardeSic and J. Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (197 I) 31-59.
S. MardeSii- and J. Segal. Equivalence of the Borsuk and the ANR-system approach to shapes. Fund. Math. 72 (1971) 61-68.
S. MardeSiC and J. Segal. Shape Theory (North-Holland. Amsterdam, 1982).
M.A. Moron and F.R. Ruiz del Portal. Multivalued maps and shape for paracompacta. Math. Japon. 39 (1994) 489-500.
M.A. Moron and F.R. Ruiz de1 Portal, Shape as a Cantor completion process, Math. Z., to appear.
P. Mrozik. Some applications of lattice theory to shape theory. Arch. Math. 47 ( 1986) 243-250.
J.M.R. Sanjurjo. A non-continuous description of the shape category of compacta, Quart. J. Math. Oxford (2) 40 (1989) 35 1-359.
J.M.R. Sanjurjo, Multihomotopy sets and transformations induced by shape, Quart, J. Math. Oxford (2) 42 ( 1991) 489-499.
J.M.R. Sanjurjo. An intrinsic description of shape. Trans. Amer. Math. Sot. 329 (1992) 625-636.
J.M.R. Sanjurjo. Multihomotopy, Tech spaces of loops and shape groups, Proc. London Math. Sot. (3) 69 (1994) 330-344.
J.M.R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl. I52 (1995) 519-528.
P. Saperstone. Semidynamical Systems in Infinite Dimensional Spaces. Appl. Math. Sci. 37 (Springer. Berlin. I98 I ).
|Deposited On:||31 Oct 2012 10:44|
|Last Modified:||07 Feb 2014 09:39|
Repository Staff Only: item control page