Geodesic
Monomorphisms in spaces with Lindelöf filters
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 281-317 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

$\mathbf{SpFi}$ is the category of spaces with filters: an object is a pair $(X,\mathcal{F}) $, $X$ a compact Hausdorff space and $\mathcal{F}$ a filter of dense open subsets of $X$. A morphism $f\: (Y,\mathcal{G}) \rightarrow (X,\mathcal{F}) $ is a continuous function $f\: Y\rightarrow X$ for which $f^{-1}(F) \in \mathcal{G}$ whenever $F\in \mathcal{F}$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in $\mathbf{SpFi}$. Of course, these monomorphisms need not be one-to-one. For general $\mathbf{SpFi}$ we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category $\mathbf{LSpFi}$ of spaces with Lindelöf filters, meaning filters with a base of Lindelöf, or cozero, sets, the criterion becomes a real characterization with several foci ($C(X) $, Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given $(X,\mathcal{F}) \in \mathbf{LSpFi}$. Considerable attempt is made to keep the discussion “topological,” i.e., within $\mathbf{SpFi}$, and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean $\ell $-groups with unit, roughly dual to monics in $\mathbf{LSpFi}$.
$\mathbf{SpFi}$ is the category of spaces with filters: an object is a pair $(X,\mathcal{F}) $, $X$ a compact Hausdorff space and $\mathcal{F}$ a filter of dense open subsets of $X$. A morphism $f\: (Y,\mathcal{G}) \rightarrow (X,\mathcal{F}) $ is a continuous function $f\: Y\rightarrow X$ for which $f^{-1}(F) \in \mathcal{G}$ whenever $F\in \mathcal{F}$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in $\mathbf{SpFi}$. Of course, these monomorphisms need not be one-to-one. For general $\mathbf{SpFi}$ we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category $\mathbf{LSpFi}$ of spaces with Lindelöf filters, meaning filters with a base of Lindelöf, or cozero, sets, the criterion becomes a real characterization with several foci ($C(X) $, Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given $(X,\mathcal{F}) \in \mathbf{LSpFi}$. Considerable attempt is made to keep the discussion “topological,” i.e., within $\mathbf{SpFi}$, and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean $\ell $-groups with unit, roughly dual to monics in $\mathbf{LSpFi}$.
Classification : 05A15, 05C38, 15A15, 15A18, 18A20, 54D30
Keywords: compact Hausdorff space; Lindelöf set; monomorphism 
@article{CMJ_2007_57_1_a22,
     author = {Ball, Richard N. and Hager, Anthony W.},
     title = {Monomorphisms in spaces with {Lindel\"of} filters},
     journal = {Czechoslovak Mathematical Journal},
     pages = {281--317},
     year = {2007},
     volume = {57},
     number = {1},
     mrnumber = {2309966},
     zbl = {1174.05066},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a22/}
}
TY  - JOUR
AU  - Ball, Richard N.
AU  - Hager, Anthony W.
TI  - Monomorphisms in spaces with Lindelöf filters
JO  - Czechoslovak Mathematical Journal
PY  - 2007
SP  - 281
EP  - 317
VL  - 57
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a22/
LA  - en
ID  - CMJ_2007_57_1_a22
ER  - 
%0 Journal Article
%A Ball, Richard N.
%A Hager, Anthony W.
%T Monomorphisms in spaces with Lindelöf filters
%J Czechoslovak Mathematical Journal
%D 2007
%P 281-317
%V 57
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a22/
%G en
%F CMJ_2007_57_1_a22
Ball, Richard N.; Hager, Anthony W. Monomorphisms in spaces with Lindelöf filters. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 281-317. http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a22/

[1] R. Ball and A. Hager: Archimedean kernel-distinguishing extensions of archimedean $\ell $-groups with weak unit. Indian J. Math. 29 (1987), 351–368. | MR

[2] R. Ball and A. Hager: Applications of spaces with filters to archimedean $\ell $-groups with weak unit. Ordered Algebraic Structures, J. Martinez (ed.), Kluwer, Dordrecht, 1989, pp. 99–112. | MR

[3] R. Ball and A. Hager: Characterization of epimorphisms in archimedean lattice-ordered groups and vector lattices. Lattice-Ordered Groups, Advances and Techniques, A. Glass and C. Holland (eds.), Kluwer, Dordrecht, 1989, pp. 175–205. | MR

[4] R. Ball and A. Hager: Epicomplete archimedean $\ell $-groups. Trans. Amer. Math. Soc. 322 (1990), 459–478. | MR

[5] R. Ball and A. Hager: Epicompletion of archimedean $\ell $-groups and vector lattices with weak unit. J. Austral. Math. Soc. (1990), 25–56. | MR

[6] R. Ball, A. Hager and A. Macula: An $\alpha $-disconnected space has no proper monic preimage. Top. Appl. 37 (1990), 141–151. | MR

[7] R. Ball, A. Hager and A. Molitor: Spaces with filters. Proc. Symp. Cat. Top. Univ. Cape Town 1994, C. Gilmour, B. Banaschewski and H. Herrlich (eds.), Dept. Math. and Appl. Math., Univ. Cape Town, 1999, pp. 21–36. | MR

[8] R. Ball, A. Hager and C. NeVille: The quasi-$F_\kappa $ cover of a compact Hausdorff space and the $\kappa $-ideal completion of an archimedean $\ell $-group. General Topology and its Applications, R. M. Shortt (ed.), Dekker Notes 123, Marcel Dekker, 1989, pp. 1–40.

[9] R. Ball and J. Walters-Wayland: $C$- and $C^*$-quotients in pointfree topology. Dissertationes Mathematicae 412, Warszawa, 2002. | MR

[10] W. W. Comfort and A. Hager: Estimates for the number of real-valued continuous functions. Trans. Amer. Math. Soc. 150 (1970), 618–631. | MR

[11] R. Engelking: General Topology. Revised and completed ed., Sigma Series in Pure Mathematics; Vol. 6, Heldermann, Berlin, 1989. | MR | Zbl

[12] Z. Frolík: On bianalytic spaces. Czech. Math. J. 88 (1963), 561–573.

[13] L. Gillman and M. Jerison: Rings of Continuous Functions. Van Nostrand, Princeton, 1960, reprinted as Springer-Verlag Graduate Texts 43, Berlin-Heidelberg-New York, 1976. | MR

[14] A. Hager: Monomorphisms in Spaces with Filters. Lecture at Curacao Math. Foundation Conference on Locales and Topological Groups, 1989.

[15] A. Hager and L. Robertson: Representing and ringifying a Riesz space. Sympos. Math. XXI (1977), 411–431. | MR

[16] P. Halmos: Lectures on Boolean Algebras. Van Nostrand, 1963. | MR | Zbl

[17] M. Henriksen and D. Johnson: The structure of a class of archimedean lattice-ordered algebras. Fund. Math. 50 (1961/1962), 73–94. | MR

[18] H. Herrlich and G. Strecker: Category Theory. Allyn and Bacon, Inc., 1973. | MR

[19] P. Johnstone: Stone Spaces. Cambridge University Press, Cambridge, 1982. | MR | Zbl

[20] S. Koppelberg: Handbook of Boolean Algebras, I. J. Monk, with R. Bonnet (ed.), North Holland, Amsterdam, 1989. | MR

[21] A. Macula: Archimedean vector lattices versus topological spaces with filters. Ph.D. thesis, Wesleyan University, 1989.

[22] A. Macula: Monic sometimes means $\alpha $-irreducible. General Topology and its Applications, S. J. Andima et al. (eds.), Dekker Notes 134, Marcel Dekker, 1991, pp. 239–260. | MR | Zbl

[23] J. Madden and J. Vermeer: Epicomplete archimedean $\ell $-groups via a localic Yosida theorem. J. Pure Appl. Algebra 68 (1990), 243–252. | DOI | MR

[24] J. Madden: $\kappa $-frames. J. Pure Appl. Algebra 70 (1991), 107–127. | DOI | MR | Zbl

[25] J. Madden and A. Molitor: Epimorphisms of frames. J. Pure Appl. Algebra 70 (1991), 129–132. | DOI | MR

[26] R. D. Mauldin: Baire functions, Borel sets, and ordinary function systems. Adv. Math. 12 (1974), 418–450. | DOI | MR | Zbl

[27] J. Mioduszewski and L. Rudolph: $H$-closed and extremally disconnected Hausdorff spaces. Diss. Math. 66 (1969). | MR

[28] A. Molitor: Covers of compact Hausdorff spaces via localic methods. Ph.D. thesis, Wesleyan University, 1992.

[29] A. Molitor: A localic construction of some covers of compact Hausdorff spaces. General Topology and Applications, R. M. Shortt (ed.), Dekker Notes 123, Marcel Dekker, 1990, pp. 219–226. | MR | Zbl

[30] Z. Semadeni: Banach Spaces of Continuous Functions. PWN Publishers, 1971. | MR | Zbl

[31] R. Sikorski: Boolean Algebras, third ed. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1969.

[32] M. Stone: Boundedness properties in function lattices. Canad. J. Math. 1 (1949), 176–186. | DOI | MR | Zbl

[33] Z. Tzeng: Extended real-valued functions and the projective resolution of a compact Hausdorff space. Ph.D. thesis, Wesleyan Univ., 1970.

[34] J. Vermeer: The smallest basically disconnected preimage of a space. Topology Appl. 17 (1984), 217–232. | DOI | MR | Zbl

[35] R. Woods: Covering properties and coreflective subcategories. Proc. CCNY Conference on Limits 1987, Ann. NY Acad. Sc. 552, 173–184. | MR | Zbl

[36] K. Yosida: On the representation of the vector lattice. Proc. Imp. Acad. Tokyo 18 (1942), 339–342. | MR | Zbl