# Article

Full entry | PDF   (0.2 MB)
Keywords:
function spaces; topology of pointwise convergence; tightness
Summary:
We prove that if there is an open mapping from a subspace of \$C_p(X)\$ onto \$C_p(Y)\$, then \$Y\$ is a countable union of images of closed subspaces of finite powers of \$X\$ under finite-valued upper semicontinuous mappings. This allows, in particular, to prove that if \$X\$ and \$Y\$ are \$t\$-equivalent compact spaces, then \$X\$ and \$Y\$ have the same tightness, and that, assuming \$2^{\frak t}>\frak c\$, if \$X\$ and \$Y\$ are \$t\$-equivalent compact spaces and \$X\$ is sequential, then \$Y\$ is sequential.
References:
[Arh1] Arhangel'skii A.V.: The spectrum of frequencies of a topological space and the product operation. Trudy Moskov. Mat. Obshch. 40 (1979), 171-206 Russian English translation: Trans. Moscow Math. Soc. (1981), 40 2 169-199. MR 0550259
[Arh2] Arhangel'skii A.V.: Problems in \$C_p\$-theory. 603-615 Open Problems in Topology J. van Mill and G.M. Reed North-Holland (1990).
[Arh3] Arhangel'skii A.V.: Topological Function Spaces. Kluwer Acad. Publ. Dordrecht (1992). MR 1485266
[vDo] van Douwen E.K.: The Integers and Topology. 111-167 Handbook of Set-Theoretic Topology K. Kunen and J.E. Vaughan North-Holland Amsterdam (1984). MR 0776622 | Zbl 0561.54004
[Eng] Engelking R.: General Topology. PWN (1977), Warszawa. MR 0500780 | Zbl 0373.54002
[GH] Gul'ko S.P., Khmyleva T.E.: Compactness is not preserved by the relation of \$t\$-equivalence. Matematicheskie Zametki 39 6 (1986), 895-903 Russian English translation: Math. Notes 39 5-6 (1986), 484-488. MR 0855937
[Mal] Malykhin V.I.: On tightness and the Suslin number in \$\exp X\$ and in a product of spaces. Dokl. Akad. Nauk SSSR 203 (1972), 1001-1003 Russian English translation: Soviet Math. Dokl. (1972), 13 496-499. MR 0300241
[Ok1] Okunev O.: Weak topology of a dual space and a \$t\$-equivalence relation. Matematicheskie Zametki 46 1 53-59 (1989), Russian English translation: Math. Notes 46 1-2 534-536 (1989). MR 1019256
[Ok2] Okunev O.: A method for constructing examples of \$M\$-equivalent spaces. Topology Appl. 36 157-171 (1990), Correction Topology Appl. 49 191-192 (1993). MR 1068167 | Zbl 0779.54008
[Ra] Ranchin D.: Tightness, sequentiality and closed coverings. Dokl. AN SSSR 32 (1977), 1015-1018 Russian English translation: Soviet Math. Dokl. (1977), 18 1 196-199. MR 0436074 | Zbl 0371.54010
[Tk1] Tkachuk V.V.: Duality with respect to the functor \$C_p\$ and cardinal invariants of the type of the Souslin number. Matematicheskie Zametki 37 3 (1985), 441-445 Russian English translation: Math. Notes, 37 3 (1985), 247-252. MR 0790433
[Tk2] Tkachuk V.V.: Some non-multiplicative properties are \$l\$-invariant. Comment. Math. Univ. Carolinae 38 1 (1997), 169-175. MR 1455481 | Zbl 0886.54005

Partner of