Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A characterization for the products of $ k$- and $ \aleph \sb{0}$-spaces and related results


Author: Yoshio Tanaka
Journal: Proc. Amer. Math. Soc. 59 (1976), 149-155
MSC: Primary 54E35; Secondary 54B10, 54D50
DOI: https://doi.org/10.1090/S0002-9939-1976-0415580-6
MathSciNet review: 0415580
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: E. Michael introduced the notion of $ {\aleph _0}$-spaces and characterized spaces which are both $ k$-spaces and $ {\aleph _0}$-spaces (or, briefly, $ k$-and- $ {\aleph _0}$-spaces) as being precisely the quotient images of separable metric spaces.

The purpose of this paper is to give a necessary and sufficient condition for the product of two $ k$-and- $ {\aleph _0}$-spaces to be a $ k$-and- $ {\aleph _0}$-space. Moreover, as related matters, we shall consider the products of $ k$-spaces having other properties.


References [Enhancements On Off] (What's this?)

  • [1] A. V. Arhangel'skiĭ, Bicompact sets and the topology of spaces, Trudy Moskov. Mat. Obšč. 13 (1965), 3-55 = Trans. Moscow Math. Soc. 13 (1965), 1-62. MR 33 #3251. MR 0195046 (33:3251)
  • [2] C. J. R. Borges, On stratifiable spaces, Pacific J. Math. 17 (1966), 1-16. MR 32 #6409. MR 0188982 (32:6409)
  • [3] D. E. Cohen, Spaces with weak topology, Quart. J. Math. Oxford Ser. (2) 5 (1954), 77-80. MR 16, 62. MR 0063043 (16:62c)
  • [4] S. P. Franklin, Spaces in which sequences suffice, Fund. Math. 57 (1965), 107-115. MR 31 #5184. MR 0180954 (31:5184)
  • [5] E. Michael, Another note on paracompact spaces, Proc. Amer. Math. Soc. 8 (1957), 822-828. MR 19, 299. MR 0087079 (19:299c)
  • [6] -, A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173-176. MR 31 #1659. MR 0177396 (31:1659)
  • [7] -, $ {\aleph _0}$-spaces, J. Math. Mech. 15 (1966), 983-1002. MR 34 #6723. MR 0206907 (34:6723)
  • [8] -, Bi-quotient maps and Cartesian products of quotient maps, Ann. Inst. Fourier (Grenoble) 18 (1968), fasc. 2, 287-302. MR 39 #6277. MR 0244964 (39:6277)
  • [9] -, A quintuple quotient quest, General Topology and Appl. 2 (1972), 91-138. MR 46 #8156. MR 0309045 (46:8156)
  • [10] J. Milnor, Construction of universal bundles. I, Ann. of Math. (2) 63 (1956), 272-284. MR 17, 994. MR 0077122 (17:994b)
  • [11] K. Morita, On decomposition spaces of locally compact spaces, Proc. Japan Acad. 32 (1956), 544-548. MR 19, 49. MR 0085495 (19:49d)
  • [12] -, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365-382. MR 29 #2773. MR 0165491 (29:2773)
  • [13] K. Nagami, Dimension for $ \sigma $-metric spaces, J. Math. Soc. Japan 23 (1971), 123-129. MR 44 #4725. MR 0287521 (44:4725)
  • [14] -, $ \Sigma $-spaces, Fund. Math. 65 (1969), 170-192. MR 41 #2612. MR 0257963 (41:2612)
  • [15] A. Okuyama, Some generalizations of metric spaces, their metrization theorems and product spaces, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 9 (1968), 236-254; correction, ibid. 10 (1969), 134. MR 37 #5846; 40 #6498. MR 0230283 (37:5846)
  • [16] P. O'Meara, A metrization theorem, Math. Nachr. 45 (1970), 69-72. MR 43 #5489. MR 0279768 (43:5489)
  • [17] -, On paracompactness in function spaces with the compact-open topology, Proc. Amer. Math. Soc. 29 (1971), 183-189. MR 43 #2659. MR 0276919 (43:2659)
  • [18] F. Siwiec, Sequence-covering and countable bi-quotient mappings, General Topology and Appl. 1 (1971), no. 2, 143-154. MR 44 #5933. MR 0288737 (44:5933)
  • [19] Y. Tanaka, On quasi-$ k$-spaces, Proc. Japan Acad. 46 (1970), 1074-1079. MR 45 #5946. MR 0296887 (45:5946)
  • [20] -, Products of sequential spaces, Proc. Amer. Math. Soc. 54 (1976), 371-375. MR 0397665 (53:1523)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54E35, 54B10, 54D50

Retrieve articles in all journals with MSC: 54E35, 54B10, 54D50


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0415580-6
Keywords: Pseudobases, $ {\aleph _0}$-spaces, $ k$-networks, $ \aleph $-spaces, cosmic spaces, $ k$-spaces, strongly Fréchet spaces, $ M$-spaces, class $ \mathfrak{S}'$, class $ \mathfrak{T}'$
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society