# zbMATH — the first resource for mathematics

Spaces not distinguishing convergences of real-valued functions. (English) Zbl 1067.54028
Summary: In [Topology Appl. 41, 25–40 (1991; Zbl 0768.54025)] we have introduced the notion of a wQN-space as a space in which for every sequence of continuous functions converging pointwisely to 0 there is a subsequence converging quasi-normally to 0. In the present paper we continue this investigation and generalize some concepts touched there. The content is a variety of notions and relationships among them. The result is another scale in the investigation of smallness and the question is how this scale fits with other known scales and whether all relations in it are proper.

##### MSC:
 54G99 Peculiar topological spaces 54C30 Real-valued functions in general topology 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
Full Text:
##### References:
 [1] Bartoszyński, T., Additivity of measure implies additivity of category, Trans. amer. math. soc., 281, 209-213, (1984) · Zbl 0538.03042 [2] Bartoszyński, T.; Scheepers, M., A-sets, Real anal. exchange, 19, 521-528, (199394) · Zbl 0822.03028 [3] Bukovská, Z., Quasinormal convergence, Math. slovaca, 41, 137-146, (1991) · Zbl 0757.40004 [4] Bukovská, Z.; Bukovský, L.; Ewert, J., Quasi-uniform convergence and $$L$$-spaces, Real anal. exchange, 18, 321-329, (199293) · Zbl 0873.54018 [5] Bukovský, L.; Kholshchevnikova, N.N.; Repický, M., Thin sets of harmonic analysis and infinite combinatorics, Real anal. exchange, 20, 454-509, (199495) · Zbl 0835.42001 [6] Bukovský, L.; Recław, I.; Repický, M., Spaces not distinguishing pointwise and quasinormal convergence of real functions, Topology appl., 41, 25-40, (1991) · Zbl 0768.54025 [7] Császár, Á.; Laczkovich, M., Discrete and equal convergence, Studia sci. math. hungar., 10, 463-472, (1975) · Zbl 0405.26006 [8] Daniels, P., Pixley-roy spaces over subsets of the reals, Topology appl., 29, 93-106, (1988) · Zbl 0656.54007 [9] Denjoy, A., Leçons sur le calcul des coefficients d’une série trigonométrique, $$2\^{}\{e\}$$ partie, (1941), Paris [10] van Douwen, E.K., The integers and topology, (), 111-167 [11] Engelking, R., General topology, (1977), PWN Warszawa [12] Erdös, P.; Kunen, K.; Mauldin, R.D., Some additive properties of sets of real numbers, Fund. math., 113, 187-199, (1981) · Zbl 0482.28001 [13] Fremlin, D.H., Sequential convergence in $$Cp(X)$$, Comment. math. univ. carolin., 35, 371-382, (1994) · Zbl 0827.54002 [14] Galvin, F.; Miller, A.W., $$γ$$-sets and other singular sets of real numbers, Topology appl., 17, 145-155, (1984) · Zbl 0551.54001 [15] Iséki, K., A characterization of pseudo-compact spaces, Proc. Japan acad., 33, 320-322, (1957) · Zbl 0082.16003 [16] Iséki, K., Pseudo-compactness and $$μ$$-convergence, Proc. Japan acad., 33, 368-371, (1957) · Zbl 0082.16101 [17] Just, W.; Miller, A.W.; Scheepers, M.; Szeptycki, P.J., The combinatorics of open covers. II, Topology appl., 73, 3, 241-266, (1996) · Zbl 0870.03021 [18] Kholshchevnikova, N.N., Representation of some functions under the assumption of Martin’s axiom, Mat. zametki, Math. notes, 49, 1-2, 225-227, (1991), (in Russian); transl. in · Zbl 0729.03028 [19] Kliś, C., An example of noncomplete normed $$(K)$$-space, Bull. acad. polon. sci. ser. sci. math. astronom. phys., 26, 415-420, (1978) · Zbl 0393.46017 [20] Kuratowski, K., Topologie I, (1958), PWN Warsaw [21] Martin, A.D.; Solovay, R.M., Internal Cohen extensions, Ann. math. logic, 2, 143-178, (1970) · Zbl 0222.02075 [22] Miller, A.W., Mapping a set of reals onto the reals, J. symbolic logic, 48, 3, 575-584, (1982) · Zbl 0527.03031 [23] Miller, A.W., Special subsets of the real line, (), 201-233 [24] Recław, I., Metric spaces not distinguishing pointwise and quasinormal convergence of real functions, Bull. Polish acad. sci. math., 45, 3, 287-289, (1997) · Zbl 0897.54013 [25] M. Repický, Spaces not distinguishing convergences, Preprint, 1999 [26] Scheepers, M., Combinatorics of open covers: Ramsey theory, Topology appl., 69, 31-62, (1996) · Zbl 0848.54018 [27] Scheepers, M., $$Cp(X)$$ and arhangel’skiı̆’s $$αi$$-spaces, Topology appl., 89, 265-275, (1998) · Zbl 0930.54017 [28] M. Scheepers, Sequential convergence in $$Cp(X)$$ and property $$S1(Γ,Γ)$$, Preprint [29] Todorcevic, S., Partitions problems in topology, Contemporary mathematics, 84, (1989), Amer. Math. Soc. Providence, RI [30] Todorcevic, S., Topics in topology, Lecture notes in math., 1652, (1997), Springer Berlin · Zbl 0953.54001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.