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Keywords:
open function; cozeroset preserving function; z-open function; F-space; SV space; finite rank
open function; cozeroset preserving function; z-open function; F-space; SV space; finite rank
Summary:
A function $f$ mapping the topological space $X$ to the space $Y$ is called a {\it z-open\/} function if for every cozeroset neighborhood $H$ of a zeroset $Z$ in $X$, the image $f(H)$ is a neighborhood of $\operatorname{cl}_Y(f(Z))$ in $Y$. We say $f$ has the {\it z-separation property\/} if whenever $U$, $V$ are cozerosets and $Z$ is a zeroset of $X$ such that $U\subseteq Z\subseteq V$, there is a zeroset $Z'$ of $Y$ such that $f(U)\subseteq Z'\subseteq f(V)$. A surjective function is z-open if and only if it maps cozerosets to cozerosets and has the z-separation property. We investigate z-open functions and other functions that map cozerosets to cozerosets. We show that if $f$ is a continuous z-open function, then the Stone extension of $f$ is an open function. This is used to show several properties of topological spaces related to F-spaces are preserved under continuous z-open functions.
References:
[AP] Arkhangelskii A.V., Ponomarev V.I.: Fundamentals of General Topology. Reidel Publishing, Boston, 1983. MR 0785749
[CH] Comfort W., Hager A.: The projection mapping and other continuous functions on a product space. Math. Scand. (1971), 28 3 77-90. MR 0315657 | Zbl 0217.47904
[D] Dow A.: On F-spaces and F$ '$-spaces. Pacific J. Math. (1983), 108 2 275-284. MR 0713737 | Zbl 0474.54015
[DHH] Dashiell F., Hager A., Henriksen M.: Order-Cauchy completions of rings and vector lattices of continuous functions. Canad. J. Math. (1980), 32 3 657-685. MR 0586984 | Zbl 0462.54009
[GH] Gillman L., Henriksen M.: Rings of continuous functions in which every finitely generated ideal is principal. Trans. Amer. Math. Soc. (1956), 82 2 366-391. MR 0078980 | Zbl 0073.09201
[GJ] Gillman L., Jerison M.: Rings of Continuous Functions. D. Van Nostrand Publishing, New York, 1960. MR 0116199 | Zbl 0327.46040
[H] Hindman N.: The product of F-spaces with P-spaces. Pacific J. Math. (1973), 47 473-480. MR 0331309 | Zbl 0237.54029
[HLMW] Henriksen M., Larson S., Martinez J., Woods R.G.: Lattice-ordered algebras that are subdirect products of valuation domains. Trans. Amer. Math. Soc. (1994), 345 193-221. MR 1239640 | Zbl 0817.06014
[HVW] Henriksen M., Vermeer H., Woods R.G.: Quasi F-covers of Tychonoff spaces. Trans. Amer. Math. Soc. (1987), 303 2 779-803. MR 0902798 | Zbl 0653.54025
[HW] Henriksen M., Woods R.G.: Cozero complemented spaces; When the space of minimal prime ideals of a $C(X)$ is compact. Topology Appl. (2004), 141 147-170. MR 2058685 | Zbl 1067.54015
[HW1] Henriksen M., Wilson R.: When is $C(X)/P$ a valuation ring for every prime ideal $P$?. Topology Appl. (1992), 44 175-180. MR 1173255 | Zbl 0801.54014
[HW2] Henriksen M., Wilson R.: Almost discrete SV-spaces. Topology Appl. (1992), 46 89-97. MR 1184107
[I] Isiwata T.: Mappings and spaces. Pacific J. Math. (1967), 20 3 455-480. MR 0219044 | Zbl 0149.40501
[L1] Larson S.: Convexity conditions on $f$-rings. Canadian J. Math. (1986), 38 48-64. MR 0835035 | Zbl 0588.06011
[L2] Larson S.: $f$-Rings in which every maximal ideal contains finitely many minimal prime ideals. Comm. Algebra (1997), 25 12 3859-3888. MR 1481572 | Zbl 0952.06026
[L3] Larson S.: Constructing rings of continuous functions in which there are many maximal ideals with nontrivial rank. Comm. Algebra (2003), 31 5 2183-2206. MR 1976272 | Zbl 1024.54015
[L4] Larson S.: Rings of continuous functions on spaces of finite rank and the SV property. Comm. Algebra, to appear. MR 2345805 | Zbl 1146.54008
[L5] Larson S.: Images and open subspaces of SV spaces. Comm. Algebra, to appear. MR 2387527 | Zbl 1147.54008
[MW] Martinez J, Woodward S.: Bezout and Prüfer $f$-rings. Comm. Algebra (1992), 20 2975-2989. MR 1179272 | Zbl 0766.06018
[W] Weir M.: Hewitt-Nachbin Spaces. North Holland, Amsterdam, 1975. MR 0514909 | Zbl 0314.54002
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