Stone-Weierstrass theorems for with the sequential topology

Author:
Zdeněk Frolík

Journal:
Proc. Amer. Math. Soc. **27** (1971), 486-494

MSC:
Primary 54.40

MathSciNet review:
0270337

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Abstract: The spaces with one of the following two properties are studied: every continuous function is a Baire function with respect to any algebra of continuous functions such that projectively generates the topology, or with respect to any algebra which distinguishes the points. The former property is equivalent to the statement that of any pair of disjoint zero sets at least one is Lindelöf, the latter implies that the space is Lindelöf and is implied by analyticity. Connections with the Blackwell problem are shown.

**[1]**Eduard Čech,*Topological spaces*, Revised edition by Zdeněk Frolík and Miroslav Katětov. Scientific editor, Vlastimil Pták. Editor of the English translation, Charles O. Junge, Publishing House of the Czechoslovak Academy of Sciences, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966. MR**0211373****[2]**Zdeněk Frolík,*A measurable map with analytic domain and metrizable range is quotient.*, Bull. Amer. Math. Soc.**76**(1970), 1112–1117. MR**0265539**, 10.1090/S0002-9904-1970-12584-8**[3]**Zdeněk Frolík,*A survey of separable descriptive theory of sets and spaces*, Czechoslovak Math. J.**20 (95)**(1970), 406–467. MR**0266757****[4]**Z. Frolík,*A contribution to the descriptive theory of sets and spaces*, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961) Academic Press, New York; Publ. House Czech. Acad. Sci., Prague, 1962, pp. 157–173. MR**0145471****[5]**Anthony W. Hager,*Approximation of real continuous functions on Lindelöf spaces*, Proc. Amer. Math. Soc.**22**(1969), 156–163. MR**0244748**, 10.1090/S0002-9939-1969-0244748-3**[6]**Anthony W. Hager and Donald G. Johnson,*A note on certain subalgebras of 𝐶(𝔛)*, Canad. J. Math.**20**(1968), 389–393. MR**0222647****[7]**Edwin Hewitt,*Certain generalizations of the Weierstrass approximation theorem*, Duke Math. J.**14**(1947), 419–427. MR**0021662**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1971-0270337-X

Keywords:
Lindelöf,
almost Lindelöf,
Baire-minimal,
Blackwell space,
analytic,
algebra of continuous functions,
Baire set,
Baire function,
sequential topology

Article copyright:
© Copyright 1971
American Mathematical Society