Article
MSC:
46E25,
54C30,
54C40,
54D35,
54D60 | MR 3513448 | Zbl 06604505 | DOI: 10.14712/1213-7243.2015.161
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
zero-dimensional space; strongly zero-dimensional space; $\mathbb{N}$-compact space; Banaschewski compactification; character; ring homomorphism; functionally countable subring; functional separability
zero-dimensional space; strongly zero-dimensional space; $\mathbb{N}$-compact space; Banaschewski compactification; character; ring homomorphism; functionally countable subring; functional separability
Summary:
In this short article we answer the question posed in Ghadermazi M., Karamzadeh O.A.S., Namdari M., On the functionally countable subalgebra of $C(X)$, Rend. Sem. Mat. Univ. Padova 129 (2013), 47--69. It is shown that $C_c(X)$ is isomorphic to some ring of continuous functions if and only if $\upsilon_0 X$ is functionally countable. For a strongly zero-dimensional space $X$, this is equivalent to say that $X$ is functionally countable. Hence for every $P$-space it is equivalent to pseudo-$\aleph_0$-compactness.
References:
[1] Barr M., Kennison F., Raphael R.: Searching for absolute $\mathcal{CR}$-epic spaces. Canad. J. Math. 59 (2007), 465–487. DOI 10.4153/CJM-2007-020-9 | MR 2319155
[2] Barr M., Burgess W.D., Raphael R.: Ring epimorphisms and $C(X)$. Theory Appl. Categ. 11 (2003), no. 12, 283–308. MR 1988400 | Zbl 1042.54007
[3] Burgess W.D., Raphael R.: Compactifications, $C(X)$ and ring epimorphisms. Theory Appl. Categ. 16 (2006), no. 21, 558–584. MR 2259263 | Zbl 1115.18001
[4] Bhattacharjee P., Knox M.L., McGovern W.W.: The classical ring of quotients of $C_c(X)$. Appl. Gen. Topol. 15 (2014), no. 2, 147–154. DOI 10.4995/agt.2014.3181 | MR 3267269 | Zbl 1305.54030
[5] Boulabiar K.: Characters on $C(X)$. Canad. Math. Bull. 58 (2015), 7–8. DOI 10.4153/CMB-2014-024-3 | MR 3303202 | Zbl 1312.54006
[6] Engelking R.: General Topology. PWN-Polish Sci. Publ., Warsaw, 1977. MR 0500780 | Zbl 0684.54001
[7] Engelking R., Mrówka S.: On $E$-compact spaces. Bull. Acad. Polon. Sci. 6 (1958), 429-436. MR 0097042 | Zbl 0083.17402
[8] Ercan Z., Onal S.: A remark on the homomorphism on $C(X)$. Proc. Amer. Math. Soc. 133 (2005), 3609–3611. DOI 10.1090/S0002-9939-05-07930-X | MR 2163596 | Zbl 1087.46038
[9] Ghadermazi M., Karamzadeh O.A.S., Namdari M.: On the functionally countable subalgebra of $C(X)$. Rend. Sem. Mat. Univ. Padova 129 (2013), 47–69. DOI 10.4171/RSMUP/129-4 | MR 3090630 | Zbl 1279.54015
[10] Gillman L., Jerison M.: Rings of Continuous Functions. Graduate Texts in Mathematics, 43, Springer, New York-Heidelberg, 1976. MR 0407579 | Zbl 0327.46040
[11] Hager A., Kimber C., McGovern W.W.: Unique $a$-closure for some $\ell$-groups of rational valued functions. Czechoslovak Math. J. 55 (2005), 409–421. DOI 10.1007/s10587-005-0031-z | MR 2137147 | Zbl 1081.06020
[12] Hušek M., Pulgarín A.: $C(X)$ as a real $\ell$-group. Topology Appl. 157 (2010), 1454–1459. MR 2610454 | Zbl 1195.46024
[13] Levy R., Rice M.D.: Normal $P$-spaces and the $G_{\delta}$-topology. Colloq. Math. 44 (1981), 227–240. DOI 10.4064/cm-44-2-227-240 | MR 0652582 | Zbl 0496.54034
[14] Levy R., Matveev M.: Functional separability. Comment. Math. Univ. Carolin. 51 (2010), no. 4, 705–711. MR 2858271 | Zbl 1224.54063
[15] Mrówka S.: Structures of continuous functions III. Rings and lattices of integer-valued continuous functions. Vehr. Nederl. Akad. Wetensch. Sect. I 68 (1965), 74–82. DOI 10.1016/S1385-7258(65)50008-1 | MR 0237580 | Zbl 0139.07404
[16] Mrówka S., Shore S.D.: Structures of continuous functions V. On homomorphisms of structures of continuous functions with zero-dimensional compact domain. Vehr. Nederl. Akad. Wetensch. Sect. I 68 (1965), 92–94. MR 0237582
[17] Mrówka S.: On $E$-compact spaces II. Bull. Acad. Polon. Sci. 14 (1966), 597–605. Zbl 0161.19603
[18] Mrówka S.: Further results on $E$-compact spaces I. Acta. Math. 120 (1968), 161–185. DOI 10.1007/BF02394609 | MR 0226576 | Zbl 0179.51202
[19] Mrówka S.: Structures of continuous functions I. Acta. Math. Acad. Sci. Hungar. 21 (1970), 239–259. DOI 10.1007/BF01894771 | MR 0269706 | Zbl 0229.46027
[20] Nyikos P.: Not every 0-dimensional realcompact space is $\mathbb{N}$-compact. Bull. Amer. Math. Soc 77 (1971), 392–396. DOI 10.1090/S0002-9904-1971-12709-X | MR 0282336
[21] Pelczyński A., Semadeni Z.: Spaces of continuous functions III. Spaces $C(\Omega)$ for $\Omega$ without perfect subsets. Studia Math. 18 (1959), 211-222. DOI 10.4064/sm-18-2-211-222 | MR 0107806
[22] Porter J.R., Woods R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York, 1988. MR 0918341 | Zbl 0652.54016
[23] Rudin W.: Continuous functions on compact spaces without perfect subsets. Proc. Amer. Math. Soc. 8 (1957), 39–42. DOI 10.1090/S0002-9939-1957-0085475-7 | MR 0085475 | Zbl 0077.31103
Search
Partner of
