The ε-approximated complete invariance property




complete invariance property (CIP), set of fixed points, Peano continua, α-dense curves, degree of nondensifiability


In the present paper we introduce a generalization of the complete invariance property (CIP) for metric spaces, which we will call the
ε-approximated complete invariance property (ε-ACIP). For our goals, we will use the so called degree of nondensifiability
(DND) which, roughly speaking, measures (in the specified sense) the distance from a bounded metric space to its class of Peano continua.
Our main result relates the ε-ACIP with the DND and, in particular, proves that a densifiable metric space has the ε-ACIP for each ε>0.
Also, some essentials differences between the CIP and the ε-ACIP are shown.


Download data is not yet available.

Author Biography

Gonzalo García, UNED

Departamento de Matemáticas


Y. Cherruault and G. Mora, Optimisation Globale. Théorie des Courbes α-denses, Económica, Paris, 2005.

R. Dubey and A. Vyas, Wavelets and the complete invariance property, Mat. Vesnik, 62 (2010), 183-188.

G. García and G. Mora, A fixed point result in Banach algebras based on the degree of nondensifiability and applications to quadratic integral equations, J. Math. Anal. Appl. 472 (2019), 1220-1235.

G. García and G. Mora, The degree of convex nondensifiability in Banach spaces, J. Convex Anal. 22 (2015), 871-888.

K. H. Heinrich and J. R. Martin, G-spaces and fixed point sets, Geom. Dedicata 83 (2000), 39-61.

J. R. Martin, Fixed point sets of metric and nonmetric spaces, Trans. Amer. Math. Soc. 284 (1984), 337-353.

J. R. Martin, Fixed point sets of LC∞,C∞ continua, Proc. Amer. Math. Soc. 81 (1981), 325-328.

J. R. Martin, Fixed point sets of Peano continua, Pacific J. Math. 74 (1978), 163-166.

J. R. Martin and S. B. Nadler, Examples and questions in the theory of fixed point sets, Canad. J. Math. 31 (1979), 1017-1032.

J. R. Martin and E. D. Tymchatyn, Fixed point sets of 1-dimensional Peano Continua, Pacific J. Math. 89 (1980), 147-149.

D. Masood and P. Singh, Complete invariance property on hyperspaces, JP J. Geom. Topol. 17 (2015), 83-94.

D. Masood and P. Singh, On equivariant complete invariance property, Sci. Math. Jpn. 77 (2013), 1-6.

S. C. Maury, Hyperspaces and the S-equivariant complete invariance property, Kyungpook Math. J. 55 (2015), 219-224.

G. Mora, The Peano curves as limit of α-dense curves, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 9 (2005), 23-28.

G. Mora and Y. Cherruault, Characterization and generation of α-dense curves, Computers Math. Applic. 33 (1997), 83-91.

G. Mora and J. A. Mira, Alpha-dense curves in infinite dimensional spaces, Int. J. Pure Appl. Math. 5 (2003), 437-449.

G. Mora and D. A. Redtwitz, Densifiable metric spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 105 (2011), 71-83.

M. Rahal, R. Ziadi and A. Ellaia, Generating α-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization, Croatian Operational Research Review 10 (2019), 289-314.

H. Sagan, Space-Filling Curves, Springer-Verlag, New York 1994.

L. E. Ward, Fixed point sets, Pacific J. Math. 47 (1973), 553-565.

S. Willard, General Topology, Dover Pub. Inc., New York 1970.

D. X. Zhou, Complete invariance property with respect to homeomorphism over frame multiwavelet and super-wavelet spaces, Journal of Mathematics 2014 (2014), Article ID 528342, 6 pages.




How to Cite

G. García, “The ε-approximated complete invariance property”, Appl. Gen. Topol., vol. 23, no. 2, pp. 453–462, Oct. 2022.



Regular Articles