Abstract
We investigate some properties recently studied by Cruz-Castillo, Ramı́rez-Páramo and Tenorio. These properties lie and behave in the middle ground between covering properties (in particular star corvering properties) and variations of separability. This dual perspective opens the door to numerous implications and connections among various known properties. In this paper, we present a few of them.
(Communicated by David Buhagiar)
Acknowledgement
The authors express their gratitude to the referees for their useful suggestions.
References
[1] Arhangel’skii, A. V.: Topological Functions Spaces, Kluwer Academic Publisher, 1992.Search in Google Scholar
[2] Bal, P.—Bhowmik, S.: On R-star Lindelöf spaces, Palest. J. Math. 6(1) (2017), 1–7.Search in Google Scholar
[3] Bal, P.—Bhowmik, S.—Gauld, D.: On selectively star-Lindelöf properties, J. Indian Math. Soc. 85(3–4) (2018), 291–304.10.18311/jims/2018/20145Search in Google Scholar
[4] Bella, A.: On two selection principles and the corresponding games, Topology Appl. 160 (2013), 2309–2313.10.1016/j.topol.2013.07.025Search in Google Scholar
[5] Bella, A.—Bonanzinga, M.—Matveev, M. V.: Variations of selective separability, Topology Appl. 156 (2009), 1241–1252.10.1016/j.topol.2008.12.029Search in Google Scholar
[6] Bella, A.—Bonanzinga, M.—Matveev, M. V.—Tkachuk, V. V.: Selective separability: general facts and behaviour in countable spaces, Topology Proc. 32 (2008), 15–30.Search in Google Scholar
[7] Bonanzinga, M.: Star-Lindelöf and absolutely star-Lindelöf spaces, Questions Answers Gen. Topology 16 (1998), 79–104.Search in Google Scholar
[8] Bonanzinga, M.—Cammaroto, F.—Pansera, B.—Tsaban, B.: Diagonalizations of dense families, Topology Appl. 165 (2014), 12–25.10.1016/j.topol.2014.01.001Search in Google Scholar
[9] Bonanzinga, M.—Giacopello, D.—Maesano, F.: Some properties defined by relative versions of star-covering properties II, Appl. Gen. Topol. 24(2) (2023), 391–405.10.4995/agt.2023.17926Search in Google Scholar
[10] Bonanzinga, M.— Giacopello, D.: A generalization of M-separability by networks, Atti Accad. Pelori-tana Pericolanti 101(2) (2023).Search in Google Scholar
[11] Bonanzinga, M.—Maesano, F.: Some properties defined by relative versions of star-covering properties, Topology Appl. 306(1) (2020), Art. ID 107923.10.1016/j.topol.2021.107923Search in Google Scholar
[12] Bonanzinga, M.—Cuzzupé, M. V.—Sakai, M.: On selective absolute star-Lindelöfness, Topology Appl. 221 (2017), 517–523.10.1016/j.topol.2017.02.006Search in Google Scholar
[13] Bonanzinga, M.—Matveev, M. V.—Pareek, C. M.: Some remarks on generalizations of countably compact spaces and Lindelöf spaces, Rend. Circ. Mat. Palermo (2) LI (2002), 163–174.10.1007/BF02871459Search in Google Scholar
[14] Chandra, D.—Alam, N.: Certain observations on a Ufin-type selection principle, Bull. Iranian Math. Soc. 50(1) (2024), Art. No. 9.10.1007/s41980-023-00851-ySearch in Google Scholar
[15] Cruz-Castillo, R.—Ramírez-Páramo, A.—Tenorio, J. F.: On weaker versions of R-star-Lindelöf and M-star-Lindelöf properties, Filomat 38(18) (2024).10.2298/FIL2418453CSearch in Google Scholar
[16] Cruz-Castillo, R.—Ramírez-Páramo, A.—Tenorio, J. F.: H-star Lindelöf property and weaker versions, Topology Appl. 350 (2024), Art. ID 108911.10.1016/j.topol.2024.108911Search in Google Scholar
[17] Van Douwen, E. K.: The integers and topology. In: Handbook of Set-Theoretic Topology (K. Kunen, J. E. Vaughan, eds.), Elsevier Science Publishers B. V., 1984, pp. 111–167.10.1016/B978-0-444-86580-9.50006-9Search in Google Scholar
[18] Van Douwen, E. K.—Reed, G. M.—Roscoe, A. W.—Tree, I. J.: Star covering properties, Topology Appl. 39 (1991), 71–103.10.1016/0166-8641(91)90077-YSearch in Google Scholar
[19] Engelking, R.: General Topology, 2nd ed., Sigma Ser. Pure Math., Vol. 6, Heldermann, Berlin, 1989.Search in Google Scholar
[20] Ikenaga, S.—Tani, T.: On a Topological Concept between Countable Compactness and Pseudocompactness, Research Reports of Numazu Technical College 15 (1980), 139–142.Search in Google Scholar
[21] Ikenaga, S.: Topological Concepts between “Lindelöf” and “Pseudo-Lindelöf”, Research Reports of Nara National College of Technology 26 (1990), 103–108.Search in Google Scholar
[22] Ikenaga, S.: A Class Which Contains Lindelöf Spaces, Separable Spaces and Countably Compact Spaces, Memories of Numazu College Technology 18 (1983), 105–108.Search in Google Scholar
[23] Matveev, M. V.: Absolutely countably compact spaces, Topology Appl. 58 (1994), 81–91.10.1016/0166-8641(94)90074-4Search in Google Scholar
[24] Matveev, M. V.: How weak is weak extent, Topology Appl. 119 (2002), 229–232.10.1016/S0166-8641(01)00061-XSearch in Google Scholar
[25] Miller, A.: Some properties of measure and category, Trans. Amer. Math. Soc. 266 (1981), 93–114.10.1090/S0002-9947-1981-0613787-2Search in Google Scholar
[26] Miller, A.: A characterization of the least cardinal for which the Baire category theorem fails, Proc. Amer. Math. Soc. 86(3) (1982), 498–502.10.1090/S0002-9939-1982-0671224-2Search in Google Scholar
[27] Porter, J.—Woods, R.: Extensions and Absolutes of Hausdorff Spaces, Springer, Berlin, 1988.10.1007/978-1-4612-3712-9Search in Google Scholar
[28] Sakai, M.: Property C'' and function spaces, Proc. Amer. Math. Soc. 104(3) (1988), 917–919.10.1090/S0002-9939-1988-0964873-0Search in Google Scholar
[29] Sakai, M.: Selective separability of Pixley-Roy hyperspaces, Topology Appl. 159(6) (2012), 1591–1598.10.1016/j.topol.2011.03.017Search in Google Scholar
[30] Sakai, M.: Special subsets of reals characterizing local properties of function spaces. In: Selection Principles and Covering Properties in Topology, Quaderni di Matematica, (L. D. R. Kocinac, ed.), Vol. 18, 2006, pp. 195–225.Search in Google Scholar
[31] Scheepers, M.: Combinatorics of open covers (I): Ramsey theory, Topology Appl. 69 (1996), 31–62.10.1016/0166-8641(95)00067-4Search in Google Scholar
[32] Scheepers, M.: Combinatorics of open covers (VI): Selectors for sequences of dense sets, Quaest. Math. 22 (1999), 109–130.10.1080/16073606.1999.9632063Search in Google Scholar
[33] Song, K.—Zhang, Y. Y.: Some remarks on almost Lindelöf spaces and weakly Lindelöf spaces, Mat. Vesnik 62(1) (2010), 77–83.Search in Google Scholar
[34] Tkačuk, V. V.: Remainders of discrete spaces - some applications, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (4) (1990), 18–21 (in Russian), also: Moscow Univ. Math. Bull. (in English).Search in Google Scholar
[35] Vaughan, J. E.: A countably compact, separable space which is not absolutely countably compact, Comment. Math. Univ. Carolin. 36(1) (1995), 197–201.Search in Google Scholar
[36] Vaughan, J. E.: Absolute countable compactness and property (a), 8th Topological Symposium, Prague, August 18–24, 1996 (talk).Search in Google Scholar
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Articles in the same Issue
- Weak differences, weak BCK-algebras and applications to some partial orders on rings
- Weakly κ-compact topological spaces
- On sharp radius estimates for S*(β) and a product function
- Maximal subextension and stability in m-capacity of maximal subextension of m-subharmonic functions with given boundary values
- Asymptotic behavior of fractional super-linear differential equations
- New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform
- Global dynamics of the system of difference equations
- Results on oscillatory properties of third-order functional difference equations with semi-canonical operators
- A new approach to metrical fixed point theorems
- Generalized Baker’s result and stability of functional equations using fixed point results
- Characterizations of ℕ-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
- K-theory of oriented flag manifolds
- On certain observations on split continuity and cauchy split continuity
- On the generalized eta- and theta-transformation formulas as the Hecke modular relation
- On some selective star Lindelöf-type properties
Articles in the same Issue
- Weak differences, weak BCK-algebras and applications to some partial orders on rings
- Weakly κ-compact topological spaces
- On sharp radius estimates for S*(β) and a product function
- Maximal subextension and stability in m-capacity of maximal subextension of m-subharmonic functions with given boundary values
- Asymptotic behavior of fractional super-linear differential equations
- New and improved oscillation criteria of third-order half-linear delay differential equations via canonical transform
- Global dynamics of the system of difference equations
- Results on oscillatory properties of third-order functional difference equations with semi-canonical operators
- A new approach to metrical fixed point theorems
- Generalized Baker’s result and stability of functional equations using fixed point results
- Characterizations of ℕ-compactness and realcompactness via ultrafilters in the absence of the axiom of choice
- K-theory of oriented flag manifolds
- On certain observations on split continuity and cauchy split continuity
- On the generalized eta- and theta-transformation formulas as the Hecke modular relation
- On some selective star Lindelöf-type properties