Approximation of continuous functions on pseudocompact spaces
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- by C. E. Aull PDF
- Proc. Amer. Math. Soc. 81 (1981), 490-494 Request permission
Abstract:
If ${\mathcal {S}^ * }$ is the family of subrings of ${C^ * }(X)$ such that if $S \in {\mathcal {S}^ * }$, $S$ contains the constant functions and is closed under uniform convergence, then the following are equivalent for a space $(X,\mathcal {J})$. (a) $(X,\mathcal {J})$ is pseudocompact. (b) If $S \in {\mathcal {S}^ * }$ functionally separates points and zero sets, $S$ generates $(X,\mathcal {J})$. (c) If $S \in {\mathcal {S}^ * }$ functionally separates zero sets, $S = {C^ * }(X)$. (d) If $S \in {\mathcal {S}^ * }$ generates the zero sets on $(X,\mathcal {J}),S = {C^ * }(X)$. (e) If $f \in S \in {\mathcal {S}^ * }$ and $Z(f) = \phi$ then $1/f \in S$ (even when it is required that $S$ generate the topology). (f) If $f \in S \in \mathcal {S}$ then $\left | f \right | \in S$ (even when it is required that $S$ generate the topology).References
- C. E. Aull, Functionally regular spaces, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 4, 281–288. MR 0428268
- B. Banaschewski, On the Weierstrass-Stone approximation theorem, Fund. Math. 44 (1957), 249–252. MR 92931, DOI 10.4064/fm-44-3-249-252
- Robert L. Blair and Anthony W. Hager, Extensions of zero-sets and of real-valued functions, Math. Z. 136 (1974), 41–52. MR 385793, DOI 10.1007/BF01189255
- W. T. van Est and Hans Freudenthal, Trennung durch stetige Funktionen in topologischen Räumen, Nederrl. Akad. Wetensch. Proc. Ser. A. 54=Indagationes Math. 13 (1951), 359–368 (German). MR 0046033
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
- Anthony W. Hager, Approximation of real continuous functions on Lindelöf spaces, Proc. Amer. Math. Soc. 22 (1969), 156–163. MR 244748, DOI 10.1090/S0002-9939-1969-0244748-3
- Anthony W. Hager and Donald G. Johnson, A note on certain subalgebras of $C({\mathfrak {X}})$, Canadian J. Math. 20 (1968), 389–393. MR 222647, DOI 10.4153/CJM-1968-035-4
- Edwin Hewitt, Certain generalizations of the Weierstrass approximation theorem, Duke Math. J. 14 (1947), 419–427. MR 21662
- Takesi Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455–480; correction, ibid. 23 (1967), 630–631. MR 0219044, DOI 10.2140/pjm.1967.23.630
- R. M. Stephenson Jr., Product spaces and the Stone-Weierstrass theorem, General Topology and Appl. 3 (1973), 77–79. MR 315669
- R. M. Stephenson Jr., Product spaces for which the Stone-Weierstrass theorem holds, Proc. Amer. Math. Soc. 21 (1969), 284–288. MR 250260, DOI 10.1090/S0002-9939-1969-0250260-8
- R. M. Stephenson Jr., Spaces for which the Stone-Weierstrass theorem holds, Trans. Amer. Math. Soc. 133 (1968), 537–546. MR 227753, DOI 10.1090/S0002-9947-1968-0227753-6
- M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), no. 3, 375–481. MR 1501905, DOI 10.1090/S0002-9947-1937-1501905-7 K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen reeller Argumente, S. B. Deutsch Adad. Wiss. Berlin Kl. Math. Phys. Tech. (1885), 633-639, 789-805.
- Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0264581
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 81 (1981), 490-494
- MSC: Primary 54C40; Secondary 54D30
- DOI: https://doi.org/10.1090/S0002-9939-1981-0597669-6
- MathSciNet review: 597669