C ( X ) can sometimes determine X without X being realcompact

Melvin Henriksen; Biswajit Mitra

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 4, page 711-720
  • ISSN: 0010-2628

Abstract

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As usual C ( X ) will denote the ring of real-valued continuous functions on a Tychonoff space X . It is well-known that if X and Y are realcompact spaces such that C ( X ) and C ( Y ) are isomorphic, then X and Y are homeomorphic; that is C ( X ) determines X . The restriction to realcompact spaces stems from the fact that C ( X ) and C ( υ X ) are isomorphic, where υ X is the (Hewitt) realcompactification of X . In this note, a class of locally compact spaces X that includes properly the class of locally compact realcompact spaces is exhibited such that C ( X ) determines X . The problem of getting similar results for other restricted classes of generalized realcompact spaces is posed.

How to cite

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Henriksen, Melvin, and Mitra, Biswajit. "$C(X)$ can sometimes determine $X$ without $X$ being realcompact." Commentationes Mathematicae Universitatis Carolinae 46.4 (2005): 711-720. <http://eudml.org/doc/249565>.

@article{Henriksen2005,
abstract = {As usual $C(X)$ will denote the ring of real-valued continuous functions on a Tychonoff space $X$. It is well-known that if $X$ and $Y$ are realcompact spaces such that $C(X)$ and $C(Y)$ are isomorphic, then $X$ and $Y$ are homeomorphic; that is $C(X)$determines$X$. The restriction to realcompact spaces stems from the fact that $C(X)$ and $C(\upsilon X)$ are isomorphic, where $\upsilon X$ is the (Hewitt) realcompactification of $X$. In this note, a class of locally compact spaces $X$ that includes properly the class of locally compact realcompact spaces is exhibited such that $C(X)$ determines $X$. The problem of getting similar results for other restricted classes of generalized realcompact spaces is posed.},
author = {Henriksen, Melvin, Mitra, Biswajit},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nearly realcompact space; fast set; SRM ideal; continuous functions with pseudocompact support; locally compact; locally pseudocompact; nearly realcompact space; fast set; SRM ideal; continuous functions with pseudocompact support; locally compact},
language = {eng},
number = {4},
pages = {711-720},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$C(X)$ can sometimes determine $X$ without $X$ being realcompact},
url = {http://eudml.org/doc/249565},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Henriksen, Melvin
AU - Mitra, Biswajit
TI - $C(X)$ can sometimes determine $X$ without $X$ being realcompact
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 4
SP - 711
EP - 720
AB - As usual $C(X)$ will denote the ring of real-valued continuous functions on a Tychonoff space $X$. It is well-known that if $X$ and $Y$ are realcompact spaces such that $C(X)$ and $C(Y)$ are isomorphic, then $X$ and $Y$ are homeomorphic; that is $C(X)$determines$X$. The restriction to realcompact spaces stems from the fact that $C(X)$ and $C(\upsilon X)$ are isomorphic, where $\upsilon X$ is the (Hewitt) realcompactification of $X$. In this note, a class of locally compact spaces $X$ that includes properly the class of locally compact realcompact spaces is exhibited such that $C(X)$ determines $X$. The problem of getting similar results for other restricted classes of generalized realcompact spaces is posed.
LA - eng
KW - nearly realcompact space; fast set; SRM ideal; continuous functions with pseudocompact support; locally compact; locally pseudocompact; nearly realcompact space; fast set; SRM ideal; continuous functions with pseudocompact support; locally compact
UR - http://eudml.org/doc/249565
ER -

References

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  1. Blair R., van Douwen E., Nearly realcompact spaces, Topology Appl. 47 (1992), 209-221. (1992) Zbl0772.54021MR1192310
  2. Comfort W.W., On the Hewitt realcompactification of a product space, Trans. Amer. Math. Soc. (1968), 107-118. (1968) Zbl0157.53402MR0222846
  3. Gillman L., Jerison M., Rings of Continuous Functions, Springer, New York, 1976. Zbl0327.46040MR0407579
  4. Johnson D., Mandelker M., Functions with pseudocompact support, General Topology and Appl. 3 (1973), 331-338. (1973) Zbl0277.54009MR0331310
  5. Mandelker M., Supports of continuous functions, Trans. Amer. Math. Soc 156 (1971), 73-83. (1971) Zbl0197.48703MR0275367
  6. Misra P.R., On isomorphism theorems for C ( X ) , Acta Math. Acad. Sci. Hungar. 39 (1982), 379-380. (1982) Zbl0479.54011MR0653849
  7. Schommer J., Fast sets and nearly realcompact spaces, Houston J. Math. 20 (1994), 161-174. (1994) Zbl0802.54016MR1272569
  8. Schommer J., Swardson M.A., Almost* realcompactness, Comment. Math. Univ. Carolinae 42 (2001), 385-394. (2001) MR1832157

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