Reversible filters. (English) Zbl 1368.54007
Let \(\omega\) denote the set of all non-negative integers with the discrete topology, let \(\beta\omega\) be the Čech-Stone compactification of \(\omega\) and \(\omega^*=\beta\omega\setminus\omega\).
For a filter \(\mathcal{F}\subset\mathcal{P}(\omega)\), let \(\xi(\mathcal{F})=\omega\cup\{\mathcal{F}\}\) be the topological space in which every point of \(\omega\) is isolated and all neighborhoods of \(\mathcal{F}\) are of the form \(\{\mathcal{F}\}\cup F\), where \(F\in\mathcal{F}\). Let also \(K_\mathcal{F}=\{p\in\beta\omega:\mathcal{F}\subset p\}\).
A topological space \(X\) is called reversible if every continuous bijection from \(X\) to \(X\) is a homeomorphism, see [M. Rajagopalan and A. Wilansky, J. Aust. Math. Soc. 6, 129–138 (1966; Zbl 0151.29602)].
In this paper the authors introduce and study reversible filters: a filter \(\mathcal{F}\subset\mathcal{P}(\omega)\) is reversible if the topological space \(\xi(\mathcal{F})\) is reversible. Among other results, the authors prove the following:
Theorem 3.2. For every compact space \(X\) embeddable in \(\beta\omega\), there exists a reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\).
Theorem 3.5. For every compact, extremally disconnected space \(X\) embeddable in \(\beta\omega\), there exists a non-reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\).
Theorem 4.1. If \(X\) is a compact, extremally disconnected space that can be embedded in \(\omega^*\) as a weak \(P\)-set and \(X\) has a proper clopen subspace homeomorphic to \(X\), then there is a non-reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\) and \(K_\mathcal{F}\) is a weak \(P\)-set of \(\omega^*\).
Theorem 4.2. There exists a compact, extremally disconnected space \(X\) that can be embedded in \(\omega^*\) as a weak \(P\)-set and such that if \(\mathcal{F}\) is a filter for which \(X\) is homeomorphic to \(K_\mathcal{F}\) and \(K_\mathcal{F}\) is a weak \(P\)-set, then \(\mathcal{F}\) is reversible.
Theorem 4.4. If \(X\) is a compact, extremally disconnected space that is a continuous image of \(\omega^*\), then there exists a reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\) and \(K_\mathcal{F}\) is a weak \(P\)-set of \(\omega^*\).
In the last section of their paper the authors use Martin’s axiom to improve some of the above results by constructing filters \(\mathcal{F}\) such that \(K_\mathcal{F}\) is a \(P\)-set.
For a filter \(\mathcal{F}\subset\mathcal{P}(\omega)\), let \(\xi(\mathcal{F})=\omega\cup\{\mathcal{F}\}\) be the topological space in which every point of \(\omega\) is isolated and all neighborhoods of \(\mathcal{F}\) are of the form \(\{\mathcal{F}\}\cup F\), where \(F\in\mathcal{F}\). Let also \(K_\mathcal{F}=\{p\in\beta\omega:\mathcal{F}\subset p\}\).
A topological space \(X\) is called reversible if every continuous bijection from \(X\) to \(X\) is a homeomorphism, see [M. Rajagopalan and A. Wilansky, J. Aust. Math. Soc. 6, 129–138 (1966; Zbl 0151.29602)].
In this paper the authors introduce and study reversible filters: a filter \(\mathcal{F}\subset\mathcal{P}(\omega)\) is reversible if the topological space \(\xi(\mathcal{F})\) is reversible. Among other results, the authors prove the following:
Theorem 3.2. For every compact space \(X\) embeddable in \(\beta\omega\), there exists a reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\).
Theorem 3.5. For every compact, extremally disconnected space \(X\) embeddable in \(\beta\omega\), there exists a non-reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\).
Theorem 4.1. If \(X\) is a compact, extremally disconnected space that can be embedded in \(\omega^*\) as a weak \(P\)-set and \(X\) has a proper clopen subspace homeomorphic to \(X\), then there is a non-reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\) and \(K_\mathcal{F}\) is a weak \(P\)-set of \(\omega^*\).
Theorem 4.2. There exists a compact, extremally disconnected space \(X\) that can be embedded in \(\omega^*\) as a weak \(P\)-set and such that if \(\mathcal{F}\) is a filter for which \(X\) is homeomorphic to \(K_\mathcal{F}\) and \(K_\mathcal{F}\) is a weak \(P\)-set, then \(\mathcal{F}\) is reversible.
Theorem 4.4. If \(X\) is a compact, extremally disconnected space that is a continuous image of \(\omega^*\), then there exists a reversible filter \(\mathcal{F}\) such that \(X\) is homeomorphic to \(K_\mathcal{F}\) and \(K_\mathcal{F}\) is a weak \(P\)-set of \(\omega^*\).
In the last section of their paper the authors use Martin’s axiom to improve some of the above results by constructing filters \(\mathcal{F}\) such that \(K_\mathcal{F}\) is a \(P\)-set.
Reviewer: Ivan S. Gotchev (New Britain)
MSC:
| 54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |
| 54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |
| 54G05 | Extremally disconnected spaces, \(F\)-spaces, etc. |
Citations:
Zbl 0151.29602References:
| [1] | Bartoszyński, T.; Judah, H., Set Theory. On the Structure of the Real Line (1995), AK Peters, Ltd.: AK Peters, Ltd. Wellesley, MA, xii+546 pp · Zbl 0834.04001 |
| [3] | van Douwen, E. K., The integers and topology, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 111-167 · Zbl 0561.54004 |
| [4] | Dow, A., βN, (The work of Mary Ellen Rudin. The work of Mary Ellen Rudin, Madison, WI, 1991. The work of Mary Ellen Rudin. The work of Mary Ellen Rudin, Madison, WI, 1991, Ann. N.Y. Acad. Sci., vol. 705 (1993), New York Acad. Sci.: New York Acad. Sci. New York), 47-66 · Zbl 0830.54023 |
| [5] | Dow, A.; Gubbi, A. V.; Szymański, A., Rigid Stone spaces within ZFC, Proc. Am. Math. Soc., 102, 3, 745-748 (1988) · Zbl 0647.54033 |
| [6] | García-Ferreira, S.; Uzcátegui, C., Subsequential filters, Topol. Appl., 156, 18, 2949-2959 (2009) · Zbl 1180.54006 |
| [7] | García-Ferreira, S.; Uzcátegui, C., The degree of subsequentiality of a subsequential filter, Topol. Appl., 157, 14, 2180-2193 (2010) · Zbl 1201.54022 |
| [8] | Kunen, K., Weak P-points in \(N^\ast \), (Topology, vol. II. Proc. Fourth Colloq., Budapest, 1978. Topology, vol. II. Proc. Fourth Colloq., Budapest, 1978, Colloq. Math. Soc. János Bolyai, vol. 23 (1980), North-Holland: North-Holland Amsterdam, New York), 741-749 · Zbl 0435.54021 |
| [9] | van Mill, J., An introduction to βω, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 503-567 · Zbl 0555.54004 |
| [10] | Rajagopalan, M.; Wilansky, A., Reversible topological spaces, J. Aust. Math. Soc., 6, 129-138 (1966) · Zbl 0151.29602 |
| [11] | Simon, P., Applications of independent linked families, (Topology, Theory and Applications. Topology, Theory and Applications, Eger, 1983. Topology, Theory and Applications. Topology, Theory and Applications, Eger, 1983, Colloq. Math. Soc. János Bolyai, vol. 41 (1985), North-Holland: North-Holland Amsterdam), 561-580 · Zbl 0615.54004 |
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