A poset of topologies on the set of real numbers

Chatyrko, Vitalij A., and Hattori, Yasunao. "A poset of topologies on the set of real numbers." Commentationes Mathematicae Universitatis Carolinae 54.2 (2013): 189-196. <http://eudml.org/doc/252542>.

@article{Chatyrko2013,
abstract = {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 $(\mathbb \{R\}, \tau (A)), A\subseteq \mathbb \{R\}$.},
author = {Chatyrko, Vitalij A., Hattori, Yasunao},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Sorgenfrey line; poset of topologies on the set of real numbers; Sorgenfrey line; poset of topologies},
language = {eng},
number = {2},
pages = {189-196},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A poset of topologies on the set of real numbers},
url = {http://eudml.org/doc/252542},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Chatyrko, Vitalij A.
AU - Hattori, Yasunao
TI - A poset of topologies on the set of real numbers
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 2
SP - 189
EP - 196
AB - 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 $(\mathbb {R}, \tau (A)), A\subseteq \mathbb {R}$.
LA - eng
KW - Sorgenfrey line; poset of topologies on the set of real numbers; Sorgenfrey line; poset of topologies
UR - http://eudml.org/doc/252542
ER -