A Tychonoff almost realcompactification

Woods, R. Grant

doi:10.1090/S0002-9939-1974-0331330-4

A Tychonoff almost realcompactification
HTML articles powered by AMS MathViewer

by R. Grant Woods PDF

Proc. Amer. Math. Soc. 43 (1974), 200-208 Request permission

Abstract:

Let $X$ be a Tychonoff topological space. A Tychonoff almost realcompact space $aX$ is constructed that contains $X$ as a dense subspace and has the property that if $f:X \to Y$ is continuous and $Y$ is Tychonoff and almost realcompact, then $f$ can be extended continuously to $aX$. Several characterizations of $aX$ are given, and the relationships between $aX$, the Hewitt realcompactification $vX$, and the minimal $c$-realcompactification $uX$ are investigated. Properties of the projective covers of these spaces, and their relation to $vE(X)(E(X)$ denotes the projective cover of $X$), are discussed.

References

Nancy Dykes, Mappings and realcompact spaces, Pacific J. Math. 31 (1969), 347–358. MR 263039
Nancy Dykes, Generalizations of realcompact spaces, Pacific J. Math. 33 (1970), 571–581. MR 276928
Zdeněk Frolík, A generalization of realcompact spaces, Czechoslovak Math. J. 13(88) (1963), 127–138 (English, with Russian summary). MR 155289
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
Andrew M. Gleason, Projective topological spaces, Illinois J. Math. 2 (1958), 482–489. MR 121775
Kenneth Hardy and R. Grant Woods, On $c$-realcompact spaces and locally bounded normal functions, Pacific J. Math. 43 (1972), 647–656. MR 365497
H. Herrlich and J. van der Slot, Properties which are closely related to compactness, Nederl. Akad. Wetensch. Proc. Ser. A 70=Indag. Math. 29 (1967), 524–529. MR 0222848
C. T. Liu and G. E. Strecker, Concerning almost realcompactifications, Czechoslovak Math. J. 22(97) (1972), 181–190. MR 303507
J. E. Mack and D. G. Johnson, The Dedekind completion of $C({\scr X})$, Pacific J. Math. 20 (1967), 231–243. MR 211268
Dona Papert Strauss, Extremally disconnected spaces, Proc. Amer. Math. Soc. 18 (1967), 305–309. MR 210066, DOI 10.1090/S0002-9939-1967-0210066-0
R. Grant Woods, Co-absolutes of remainders of Stone-Čech compactifications, Pacific J. Math. 37 (1971), 545–560. MR 307179
R. Grant Woods, Ideals of pseudocompact regular closed sets and absolutes of Hewitt realcompactifications, General Topology and Appl. 2 (1972), 315–331. MR 319152