Sufficient conditions for the preservation of path-connectedness in an arbitrary metric space

Savvas Andronicou; Emmanouil Milakis

Ayuda

Sufficient conditions for the preservation of path-connectedness in an arbitrary metric space

Andronicou, Savvas ^[1] ; Milakis, Emmanouil ^[1]
1. [1] University of Cyprus
  
  University of Cyprus
  
  Chipre
Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 26, Nº. 2, 2025, págs. 703-724
Idioma: inglés
DOI: 10.4995/agt.2025.22779
Enlaces
- Texto completo
Resumen
- It is proven that if (X, d) is an arbitrary metric space and U is a path-connected subset of X with M := {xi : i ∈ {1, 2, . . . , k}} ⊂ int(U), then the property of path-connectedness is also preserved in the resulting set U \M, provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions we extend the above result in the case where the set M is countably infinite. As a consequence, these results provide path-connectedness for domains with holes.
Referencias bibliográficas
- S. Andronicou and E. Milakis, Estimates on nodal domains of solutions and their applications to free boundary problems, in preparation.
- A. M. Bruckner and B. S. Thomson, Classical Real Analysis, Second Edition, 2008.
- R. Caccioppoli, Misura e integratione sugli insiemi dimensionalmente orientati, Rend. Accad. Lincei, Cl. Sc. fis. mat. nat., Sirie VIII, fasc...
- L. A. Caffarelli and N. M. Riviere, On the rectifiability of domains with finite perimeter, Ann. Scuola Norm. Sup. Pisa 3, no 2 (1976), 177-186.
- E. De Giorgi, Su una teoria generale dell misura (r-1)-dimensionale in uno spazio ad r dimensioni, Annali di Matematica pura ed applicata,...
- R. Engelking, General Topology, Sigma series in pure mathematics, Volume 6, Heldermann 1989.
- G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, Wiley, 2nd edition, 1999.
- X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl. 133 (2007), 151-161....
- A. Lemenant and E. Milakis, A stability result for nonlinear Neumann problems in Reifenberg flat domains in R^N, Publ. Mat. 55, no. 2 (2011),...
- A. Lemenant, E. Milakis, and L. V. Spinolo, Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains, J. Funct....
- Y. Y. Li and X. Yan, Anisotropic Caffarelli-Kohn-Nirenberg type inequalities, Advances in Mathematics 419 (2023), 108958. https://doi.org/10.1016/j.aim.2023.108958
- F. Lin and Z. Lin, Boundary Harnack principle on nodal domains, Sci. China Math. 65 (2022), 2441-2458. https://doi.org/10.1007/s11425-022-2016-3
- J. F. Toland, Path-connectedness in global bifurcation theory, Electron. Res. Arch. 29 (2021), no. 6, 4199-4213. https://doi.org/10.3934/era.2021079
- R. Zhong, N. Huang, and M. Wong, Connectedness and path-connectedness of solution sets to symmetric vector equilibrium problems, Taiw. Journal...