Open this publication in new window or tab >>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 $(\mathbb R, \tau(A)), A\subseteq \mathbb R$.
Keywords
Sorgenfrey line; poset of topologies on the set of real numbers
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-108328 (URN)
2014-06-262014-06-262017-12-05Bibliographically approved