The Bolzano property and the cube-like complexes

Przemysław Tkacz



Introducing the \emph{Bolzano property}, we present a topological version of the Poincaré-Miranda theorem. One simple, and one algorithmic proof that $n$-cube-like complexes have this property are given. Moreover, we investigate under what conditions the inverse limit preserves the Bolzano property. Finally, we give a characterization of the Bolzano property for locally connected spaces.


Simplicial complex; fixed point; cube-like; Poincaré-Miranda

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P. Bohl, Uber die Bewegung eines mechanischen System in der Nahe einer Gleichgewichtslage, J. Reine Angew. Math. 127 (1904), 179-276.

L.E. Brouwer, Uber Abbildung von Mannihfaltigkeeiten, Math. Ann. 71 (1911), 97-115.

J. Dugundji and A. Granas, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, 2003.

R. Engelking, General Topology, Sigma Series in Pure Mathematics, Vol. 6, Heldermann, Berlin, 1989.

M. Kidawa and P. Tkacz, The cube-like complexes and the Poincare - Miranda theorem, Topology Appl. 196 (2015), 198-207.

W. Kulpa, The Bolzano property, Filomat 8 (1994), 81-97.

W. Kulpa, The Poincare{Miranda theorem, Amer. Math. Mon. 104 (1997), No. 6, 545-550.

D. Michalik, P. Tkacz and M. Turzanski, Cube-like complexes, Steinhaus' chains and the Poincare{Miranda theorem, J. Fixed Point Theory Appl. 18 (2016), No. 1, 117-131.

C. Miranda, Un'osservazione su una teorema di Brouwer, Boll. Un. Mat. Ital. 2 (1940), No. 3, 5-7.

H. Poincare, Sur certaines solutions particulieres du probleme des trois corps, C.R. Acad. Sci. Paris 97 (1883), 251-252.

H. Poincare, Sur certaines solutions particulieres du probleme des trois corps, Bull. Astronomique 1 (1884), 63-74.

L.A. Steen and J.A. Seebach, Jr., Counterexamples in Topology, second ed., Springer, New York, 1978.


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