Introduction to generalized topological spaces


  • Irina Zvina University of Latvia



Generalized topology, Generalized topological space, gt-space, Compatible ideal, Modulo ideal, Frame, Order generated by ideal


We introduce the notion of generalized topological space (gt-space). Generalized topology of gt-space has the structure of frame and is closed under arbitrary unions and finite intersections modulo small subsets. The family of small subsets of a gt-space forms an ideal that is compatible with the generalized topology. To support the definition of gt-space we prove the frame embedding modulo compatible ideal theorem. Weprovide some examples of gt-spaces and study key topological notions (continuity, separation axioms, cardinal invariants) in terms of generalized spaces.


Download data is not yet available.

Author Biography

Irina Zvina, University of Latvia

Department of Mathematics, Zellu str. 8, LV-1002, Riga, Latvia.


L. M. Brown and M. Diker, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets Syst. 98, no. 2 (1998), 217–224

S. Givant and P. Halmos, Introduction to Boolean Algebras, Springer Science+Business Media, LLC (2009)

T. R. Hamlett and D. Jankovic, Ideals in topological spaces and the set operator , Boll. Unione Mat. Ital., VII. Ser., B 4, no. 4 (1990), 863–874.

T. R. Hamlett and D. Jankovic, Ideals in general topology, General topology and applications, Proc. Northeast Conf., Middletown, CT (USA), Lect. Notes Pure Appl. Math.123 (1990), 115–125.

D. Jankovic and T. R. Hamlett, New topologies from old via ideals, Am. Math. Mon. 97, no.4 (1990), 295–310.

D. Jankovic and T. R. Hamlett, Compatible extensions of ideals, Boll. Unione Mat. Ital., VII. Ser., B 6, no.3 (1992), 453–465.

D. Jankovic, T. R. Hamlett and Ch. Konstadilaki, Local-to-global topological properties, Math. Jap. 52, no.1 (2000), 79–81.

I. Zvina, Complete infinitely distributive lattices as topologies modulo an ideal, Acta Univ. Latviensis, ser. Mathematics, 688 (2005), 121–128

I. Zvina, On i-topological spaces: generalization of the concept of a topological space via ideals, Appl. Gen. Topol. 7, no. 1 (2006), 51–66


How to Cite

I. Zvina, “Introduction to generalized topological spaces”, Appl. Gen. Topol., vol. 12, no. 1, pp. 49–66, Apr. 2011.



Regular Articles