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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, p. 189-196Article 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, p. 189-196
##### Keywords [en]
Sorgenfrey line; poset of topologies on the set of real numbers
Mathematics
##### Identifiers
OAI: oai:DiVA.org:liu-108328DiVA, id: diva2:729903
Available from: 2014-06-26 Created: 2014-06-26 Last updated: 2017-12-05Bibliographically approved

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#### Authority records BETA

Tjatyrko, Vitalij A.

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Mathematics and Applied MathematicsThe Institute of Technology
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Commentationes Mathematicae Universitatis Carolinae
Mathematics

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Cite
Citation style
• apa
• harvard1
• ieee
• modern-language-association-8th-edition
• vancouver
• oxford
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
• html
• text
• asciidoc
• rtf