Topologies generated by ideals

Carlos Uzcátegui

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 2, page 317-335
  • ISSN: 0010-2628

Abstract

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A topological space X is said to be generated by an ideal ℐ if for all A ⊆ X and all x ∈ A ¯ there is E ⊆ A in ℐ such that x ∈ E ¯ , and is said to be weakly generated by ℐ if whenever a subset A of X contains E ¯ for every E ⊆ A with E ∈ ℐ , then A itself is closed. An important class of examples are the so called weakly discretely generated spaces (which include sequential, scattered and compact Hausdorff spaces). Another paradigmatic example is the class of Alexandroff spaces which corresponds to spaces generated by finite sets. By considering an appropriate topology on the power set of X we show that τ is weakly generated by ℐ iff τ is a ℐ -closed subset of 𝒫 ( X ) . The class of spaces weakly generated by an ideal behaves as the class of sequential spaces, in the sense that their closure operator can be characterized as the sequential closure and moreover there is a natural notion of a convergence associated to them. We also show that the collection of topologies weakly generated by ℐ is lattice isomorphic to a lattice of pre-orders over ℐ .

How to cite

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Uzcátegui, Carlos. "Topologies generated by ideals." Commentationes Mathematicae Universitatis Carolinae 47.2 (2006): 317-335. <http://eudml.org/doc/249863>.

@article{Uzcátegui2006,
abstract = {A topological space $X$ is said to be generated by an ideal $\mathcal \{I\}$ if for all $A\subseteq X$ and all $x\in \overline\{A\}$ there is $E\subseteq A$ in $\mathcal \{I\}$ such that $x\in \overline\{E\}$, and is said to be weakly generated by$\mathcal \{I\}$ if whenever a subset $A$ of $X$ contains $\overline\{E\}$ for every $E\subseteq A$ with $E\in \mathcal \{I\}$, then $A$ itself is closed. An important class of examples are the so called weakly discretely generated spaces (which include sequential, scattered and compact Hausdorff spaces). Another paradigmatic example is the class of Alexandroff spaces which corresponds to spaces generated by finite sets. By considering an appropriate topology on the power set of $X$ we show that $\tau $ is weakly generated by $\mathcal \{I\}$ iff $\tau $ is a $\mathcal \{I\}$-closed subset of $\mathcal \{P\}(X)$. The class of spaces weakly generated by an ideal behaves as the class of sequential spaces, in the sense that their closure operator can be characterized as the sequential closure and moreover there is a natural notion of a convergence associated to them. We also show that the collection of topologies weakly generated by $\mathcal \{I\}$ is lattice isomorphic to a lattice of pre-orders over $\mathcal \{I\}$.},
author = {Uzcátegui, Carlos},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {lattices of topologies; hyperspaces; tightness; Alexandroff spaces; Fréchet and sequential spaces; discretely generated spaces; sequential convergence; lattices of topologies; hyperspaces; tightness; Alexandroff spaces; Fréchet and sequential spaces; discretely generated spaces},
language = {eng},
number = {2},
pages = {317-335},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Topologies generated by ideals},
url = {http://eudml.org/doc/249863},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Uzcátegui, Carlos
TI - Topologies generated by ideals
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 2
SP - 317
EP - 335
AB - A topological space $X$ is said to be generated by an ideal $\mathcal {I}$ if for all $A\subseteq X$ and all $x\in \overline{A}$ there is $E\subseteq A$ in $\mathcal {I}$ such that $x\in \overline{E}$, and is said to be weakly generated by$\mathcal {I}$ if whenever a subset $A$ of $X$ contains $\overline{E}$ for every $E\subseteq A$ with $E\in \mathcal {I}$, then $A$ itself is closed. An important class of examples are the so called weakly discretely generated spaces (which include sequential, scattered and compact Hausdorff spaces). Another paradigmatic example is the class of Alexandroff spaces which corresponds to spaces generated by finite sets. By considering an appropriate topology on the power set of $X$ we show that $\tau $ is weakly generated by $\mathcal {I}$ iff $\tau $ is a $\mathcal {I}$-closed subset of $\mathcal {P}(X)$. The class of spaces weakly generated by an ideal behaves as the class of sequential spaces, in the sense that their closure operator can be characterized as the sequential closure and moreover there is a natural notion of a convergence associated to them. We also show that the collection of topologies weakly generated by $\mathcal {I}$ is lattice isomorphic to a lattice of pre-orders over $\mathcal {I}$.
LA - eng
KW - lattices of topologies; hyperspaces; tightness; Alexandroff spaces; Fréchet and sequential spaces; discretely generated spaces; sequential convergence; lattices of topologies; hyperspaces; tightness; Alexandroff spaces; Fréchet and sequential spaces; discretely generated spaces
UR - http://eudml.org/doc/249863
ER -

References

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  1. Alas O.T., Tkachuk V.V., Wilson R.G., Closures of discrete sets often reflect global properties, Topology Proc. 25 27-44 (2000). (2000) Zbl1002.54021MR1875581
  2. Bella A., Simon P., Spaces which are generated by discrete sets, Topology Appl. 135 1-3 87-99 (2004). (2004) Zbl1050.54001MR2024948
  3. Bella A., Yaschenko I.V., On AP and WAP spaces, Comment. Math. Univ. Carolinae 40.3 521-536 (1999). (1999) Zbl1010.54040MR1732483
  4. Daniels P., Pixley-Roy spaces over subsets of the reals, Topology Appl. 29 93-106 (1988). (1988) Zbl0656.54007MR0944073
  5. van Douwen E.K., The Pixley-Roy topology on spaces of subsets, in: Set Theoretic Topology, G.M. Reed, editor, pp. 111-134, Academic Press, New York, 1977. Zbl0372.54006MR0440489
  6. van Douwen E.K., Applications of maximal topologies, Topology Appl. 51.2 125-139 (1993). (1993) Zbl0845.54028MR1229708
  7. Dow A., Tkachenko M.G., Tkachuk V.V., Wilson R.G., Topologies generated by discrete subspaces, Glas. Math. Ser. III 37(57) 187-210 (2002). (2002) Zbl1009.54005MR1918105
  8. Ellentuck E., A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 163-165 (1974). (1974) Zbl0292.02054MR0349393
  9. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  10. Frič R., History of sequential convergence spaces, in: Handbook of the History of General Topology, C.E. Aull and R. Lowen, editors, volume 1, pp.343-355; Kluwer Academic Publishers, Amsterdam, 1997. MR1617537
  11. Johnstone P., Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1986. Zbl0586.54001MR0861951
  12. Kašuba R., The generalized Ochan topology on sets of subsets and topological Boolean rings, Math. Nachr. 97 47-56 (1980). (1980) MR0600823
  13. Nyikos P., Convergence in topology, in: Recent Progress in General Topology, M. Hušek and J. van Mill, editors, pp.537-570; Elsevier, 1992. Zbl0794.54004MR1229138
  14. Popov V.V., Structure of the exponential of a discrete space, Mathematical Notes 35.5 (1984), 399-404; translated from the Mat. Zametki. (1984) MR0750817
  15. Steiner A.K., The lattice of topologies: structure and complementation, Trans. Amer. Math. Soc. 122.2 379-398 (1966). (1966) Zbl0139.15905MR0190893
  16. Tkachuk V.V., Yaschenko I.V., Almost closed sets and topologies they determine, Comment. Math. Univ. Carolinae 42.2 393-403 (2001). (2001) Zbl1053.54004MR1832158
  17. Todorčević S., Uzcátegui C., Analytic topologies over countable sets, Topology Appl. 111.3 299-326 (2001). (2001) Zbl0970.03042MR1814231
  18. Todorčević S., Uzcátegui C., Analytic k -spaces, Topology Appl. 146-147 511-526 (2005). (2005) Zbl1063.54015MR2107168
  19. Uzcátegui C., Vielma J., Alexandroff topologies viewed as closed sets in the Cantor cube, Divulg. Mat. 13.1 45-53 (2005). (2005) Zbl1098.54001MR2180768
  20. Watson S., The construction of topological spaces: Order, unpublished, 2001; http://math.yorku.ca/ watson/. 

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