Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
$\Sigma_s$-product; Lindelöf $\Sigma$-space; $L\Sigma(\leq \omega)$-space; monotonically monolithic space; Collins-Roscoe space; function space; simple space
$\Sigma_s$-product; Lindelöf $\Sigma$-space; $L\Sigma(\leq \omega)$-space; monotonically monolithic space; Collins-Roscoe space; function space; simple space
Summary:
We show that any $\Sigma_s$-product of at most $\mathfrak{c}$-many $L\Sigma(\leq \omega)$-spaces has the $L\Sigma(\leq \omega)$-property. This result generalizes some known results about $L\Sigma(\leq \omega)$-spaces. On the other hand, we prove that every $\Sigma_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every $\Sigma_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [{Lifting the Collins-Roscoe property by condensations}, Topology Proc. {42} (2012), 1--15]. Besides, we prove that if $X$ is a simple Lindelöf $\Sigma$-space, then $C_p(X)$ has the Collins-Roscoe property.
References:
[1] Alas O., Tkachuk V., Wilson R.: A broader context for monotonically monolithic spaces. Acta Math. Hungar. 125 (2009), no. 4, 369–385. DOI 10.1007/s10474-009-9034-9 | MR 2564435 | Zbl 1274.54043
[2] Arhangel'skii A.V.: Topological Function Spaces. Kluwer Acad. Publ., Dordrecht, 1992. MR 1485266
[3] Collins P.J., Roscoe A.W.: Criteria for metrisability. Proc. Amer. Math. Soc. 90 (1984), no. 4, 631–640. DOI 10.1090/S0002-9939-1984-0733418-9 | MR 0733418 | Zbl 0541.54034
[4] Engelking R.: General Topology. second edition, Sigma Series in Pure Mathematics, 6, Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[5] Gruenhage G.: Monotonically monolithic spaces, Corson compacts and $D$-spaces. Topology Appl. 159 (2012), 1559–1564. MR 2891421 | Zbl 1241.54013
[6] Guo H., Junnila H.J.K.: $D$-spaces and thick covers. Topology Appl. 158 (2011), 2111–2121. DOI 10.1016/j.topol.2011.06.053 | MR 2831896 | Zbl 1244.54056
[7] Kubiś W., Okunev O., Szeptycki P.J.: On some classes of Lindelöf $\Sigma$-spaces. Topology Appl. 153 (2006), 2574–2590. MR 2243735 | Zbl 1102.54028
[8] Molina Lara I., Okunev O.: $L\Sigma(\leq\omega)$-spaces and spaces of continuous functions. Central European J. Math. 8 (2010), 754–762. DOI 10.2478/s11533-010-0039-y | MR 2671223 | Zbl 1209.54005
[9] Okunev O.G.: On Lindelöf $\Sigma$-spaces of continuous functions in the pointwise topology. Topology Appl. 49 (1993), 149–166. DOI 10.1016/0166-8641(93)90041-B | MR 1206222 | Zbl 0796.54026
[10] Sokolov G.A.: On some class of compact spaces lying in $\Sigma$-products. Comment. Math. Univ. Carolin. 25 (1984), no. 2, 219–231. MR 0768809
[11] Tkačenko M.G.: $\mathcal{P}$-approximable compact spaces. Comment. Math. Univ. Carolin. 32 (1991), 583–595. MR 1159804
[12] Tkachuk V.V.: A glance at compact spaces which map “nicely” onto the metrizable ones. Topology Proc. 19 (1994), 321–334. MR 1369767 | Zbl 0854.54022
[13] Tkachuk V.V.: Behaviour of the Lindelöf $\Sigma$-property in iterated function spaces. Topology Appl. 107 (2000), 297–305. DOI 10.1016/S0166-8641(99)00112-1 | MR 1779816
[14] Tkachuk V.V.: Condensing function spaces into $\Sigma$-products of real lines. Houston J. Math. 33 (2007), no. 1, 209–228. MR 2287851 | Zbl 1126.54012
[15] Tkachuk V.V.: Lifting the Collins-Roscoe property by condensations. Topology Proc. 42 (2012), 1–15. MR 2943585 | Zbl 1280.54011
[16] Tkachuk V.V.: Monolithic spaces and $D$-spaces revisited. Topology Appl. 156 (2009), 840–846. DOI 10.1016/j.topol.2008.11.001 | MR 2492968 | Zbl 1165.54009
[17] Tkachuk V.V.: Some criteria for $C_p(X)$ to be an $L\Sigma(\leq \kappa)$-space. Rocky Mountain J. Math. 43 (2013), 373–384. DOI 10.1216/RMJ-2013-43-1-373 | MR 3065471
[18] Tkachuk V.V.: The Collins-Roscoe property and its applications in the theory of function spaces. Topology Appl. 159 (2012), 1529–1535. DOI 10.1016/j.topol.2010.08.022 | MR 2891418 | Zbl 1245.54022
Search
Partner of
