Rarely convergent sequences in topological spaces


  • Tarkan Öner Muğla Sıtkı Koçman University- Turkey


Rare sets, rare continuity, rare convergence, rarely sequential spaces


In this work, by using rare sets, we introduce a new type of convergence in topological spaces, called rare convergence, which is weaker than ordinary convergence. After presenting some examples, we investigate the relationship of rare convergence with continuity and rare continuity. Using rare convergence, we also construct a new topology, called rarely sequential topology, which is coarser than sequential topology.


Caldas, M. & Jafari, S. (2005) On rarely g-continuous functions. Glasnik matematicki, 40(2): 317-322.

Caldas, M. & Jafari, S. (2006) A strong form of rare continuity. Soochow Journal of Mathematics, 32(1): 161.

Caldas, M., Jafari, S. & Noiri, T. (2005) Characterizations of rarely g-continuous multifunctions. Sarajevo journal of mathematics,

(13): 129-133.

Ekici, E. & Jafari, S. (2009) On a new weaker form of Popa’s rare continuity via-open sets. Kuwait J. Sci. Eng., 36(1A): 33-41.

Ekici, E. & Jafari, S. (2013) On rare s-precontinuity for multifunctions. Demonstratio Mathematica, 46(2): 395-403.

Franklin, S. P. (1965) Spaces in which sequences suffice. Fundamenta Mathematicae, 57(1): 107-115.

Jafari, S. & Noiri, T. (2000) On rarely beta-continuous functions. Jour. of Inst. of Math.& Comp. Sci.(Math. Ser.), 13(2): 247-251.

Jafari, S. & Sengül, U. (2013) Rarely b-continuous functions. Italian journal of pure and applied mathematics, 30, 109-116.

Jafari, S. (2005) Rare alpha-Continuity. Bulletin of the Malaysian Mathematical Sciences Society, 28(2): 157-161.

Jafari, S. (1997) On some properties of rarely continuous functions. Univ. Bacau. Stud.Cerc. St. Ser. Mat., 7: 65-73.

Jafari, S. (1995) A note on rarely continuous functions. Univ. Bacau. Stud. Cerc. St. Ser.Mat., 5: 29-34.

Long, P.E. & Herrington, L.L. (1982) Properties of rarely continuous functions. Glasnik Mat. Ser. III, 17(37): 147-153.

Popa, V. (1989) Some properties of rarely continuous multifunction. In Conf. Nat. Geom.Topologie, Univ. Al. I. Cuza Iasi (pp. 269-274).

Popa, V. (1979) Sur certain decomposition de la continuite dans les espaces topologiques. Glasnik Mat. Ser. III, 14(34): 359-362.

Roy, B. (2014) On Unification of rarely continuous functions. Facta Universitatis, Series: Mathematics and Informatics, 29(3): 261-269.