Article
| Title: | Small objects in Top and final generation by large topologies (English) |
| Author: | Hušek, Miroslav |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 66 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 103-119 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Every reflective class of topological spaces containing all discrete spaces and being a part of $ \mathsf{Top}_1$ (or of $ \mathsf{Top}_0$ and containing the Sierpiński space) contains all the T$_2$-spaces (T$_0$-spaces, respectively) having finitely many accumulation points. As a consequence one gets characterizations of topological spaces that are small (finitely generated) in reflective categories of topological spaces. That completes and unifies results of the recent paper by J. Adámek, M. Hušek, J. Rosický and W. Tholen ``Smallness in topology'' (2023). Main tool for getting those characterizations is the fact that many spaces are finally generated by directed systems of finer spaces having finitely many accumulation points (the needed word directed makes the procedure nontrivial). (English) |
| Keyword: | final generation |
| Keyword: | finitely generated space |
| Keyword: | reflective subcategory |
| MSC: | 18F60 |
| MSC: | 54B30 |
| MSC: | 54B99 |
| DOI: | 10.14712/1213-7243.2026.003 |
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| Date available: | 2026-05-15T05:58:13Z |
| Last updated: | 2026-05-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153607 |
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| Reference: | [2] Adámek J., Hušek M., Rosický J., Tholen W.: Smallness in topology.Quest. Math. 46 (2023), suppl. 1, 13–39. 10.2989/16073606.2023.2247720 |
| Reference: | [3] Comfort W. W., Negrepontis S.: The Theory of Ultrafilters.Die Grundlehren der mathematischen Wissenschaften, 211, Springer, New York, 1974. Zbl 0298.02004 |
| Reference: | [4] Engelking R.: General Topology.Monografie Matematyczne, 60, PWN—Państwowe Wydawnictwo Naukowe, Warsaw, 1977. Zbl 0684.54001 |
| Reference: | [5] Herrlich H.: On the concept of reflections in general topology.in: Contributions to Extension Theory of Topological Structures, Proc. Sympos., Berlin, 1967, Deutscher Verlag Wissensch., Berlin, 1969, pages 105–114. Zbl 0182.25301 |
| Reference: | [6] Hewitt E.: A problem of set-theoretic topology.Duke Math. J. 10 (1943), 309–333. Zbl 0060.39407, 10.1215/S0012-7094-43-01029-4 |
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