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A poset of topologies on the set of real numbers
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Department of Mathematics, Shimane University, Matsue, Japan.
2013 (English)In: Commentationes Mathematicae Universitatis Carolinae, ISSN 0010-2628, E-ISSN 1213-7243, Vol. 54, no 2, 189-196 p.Article in journal (Refereed) Published
Abstract [en]

On the set $\mathbb R$ of real numbers we consider a poset $\mathcal P_\tau(\mathbb R)$ (by inclusion) of topologies $\tau(A)$, where $A\subseteq \mathbb R$, such that $A_1\supseteq A_2$ iff $\tau(A_1)\subseteq \tau(A_2)$. The poset has the minimal element $\tau (\mathbb R)$, the Euclidean topology, and the maximal element $\tau (\emptyset)$, the Sorgenfrey topology. We are interested when two topologies $\tau_1$ and $\tau_2$ (especially, for $\tau_2 = \tau(\emptyset)$) from the poset define homeomorphic spaces $(\mathbb R, \tau_1)$ and $(\mathbb R, \tau_2)$. In particular, we prove that for a closed subset $A$ of $\mathbb R$ the space $(\mathbb R, \tau(A))$ is homeomorphic to the Sorgenfrey line $(\mathbb R, \tau(\emptyset))$ iff $A$ is countable. We study also common properties of the spaces $(\mathbb R, \tau(A)), A\subseteq \mathbb R$.

Place, publisher, year, edition, pages
2013. Vol. 54, no 2, 189-196 p.
Keyword [en]
Sorgenfrey line; poset of topologies on the set of real numbers
National Category
URN: urn:nbn:se:liu:diva-108328OAI: diva2:729903
Available from: 2014-06-26 Created: 2014-06-26 Last updated: 2014-08-25Bibliographically approved

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Tjatyrko, Vitalij A.
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