# Article

 Title: Topologies generated by ideals  (English) Author: Uzcátegui, Carlos Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 47 Issue: 2 Year: 2006 Pages: 317-335 . Category: math . Summary: A topological space $X$ is said to be {\it generated by an ideal $\Cal I$\/} if for all $A\subseteq X$ and all $x\in \overline{A}$ there is $E\subseteq A$ in $\Cal I$ such that $x\in \overline{E}$, and is said to be {\it weakly generated by\/} $\Cal I$ if whenever a subset $A$ of $X$ contains $\overline{E}$ for every $E\subseteq A$ with $E\in \Cal I$, 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 $\Cal I$ iff $\tau$ is a $\Cal I$-closed subset of $\Cal P(X)$. 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 $\Cal I$ is lattice isomorphic to a lattice of pre-orders over $\Cal I$. Keyword: lattices of topologies Keyword: hyperspaces Keyword: tightness Keyword: Alexandroff spaces Keyword: Fréchet and sequential spaces Keyword: discretely generated spaces Keyword: sequential convergence MSC: 06B30 MSC: 54A10 MSC: 54A20 MSC: 54B20 MSC: 54D55 idZBL: Zbl 1150.54302 idMR: MR2241535 . Date available: 2009-05-05T16:57:37Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119595 . 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