When is $𝐍$ Lindelöf?

Herrlich, Horst, and Strecker, George E.. "When is $\mathbf {N}$ Lindelöf?." Commentationes Mathematicae Universitatis Carolinae 38.3 (1997): 553-556. <http://eudml.org/doc/248055>.

@article{Herrlich1997,
abstract = {Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\mathbb \{N\}$ is a Lindelöf space, (2) $\mathbb \{Q\}$ is a Lindelöf space, (3) $\mathbb \{R\}$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $\mathbb \{R\}$ is separable, (6) in $\mathbb \{R\}$, a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$, (7) a function $f:\mathbb \{R\}\rightarrow \mathbb \{R\}$ is continuous at a point $x$ iff $f$ is sequentially continuous at $x$, (8) in $\mathbb \{R\}$, every unbounded set contains a countable, unbounded set, (9) the axiom of countable choice holds for subsets of $\mathbb \{R\}$.},
author = {Herrlich, Horst, Strecker, George E.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {axiom of choice; axiom of countable choice; Lindelöf space; separable space; (sequential) continuity; (Dedekind-) finiteness; axiom of choice; axiom of countable choice; Lindelöf space; sequential continuity},
language = {eng},
number = {3},
pages = {553-556},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {When is $\mathbf \{N\}$ Lindelöf?},
url = {http://eudml.org/doc/248055},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Herrlich, Horst
AU - Strecker, George E.
TI - When is $\mathbf {N}$ Lindelöf?
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 3
SP - 553
EP - 556
AB - Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\mathbb {N}$ is a Lindelöf space, (2) $\mathbb {Q}$ is a Lindelöf space, (3) $\mathbb {R}$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $\mathbb {R}$ is separable, (6) in $\mathbb {R}$, a point $x$ is in the closure of a set $A$ iff there exists a sequence in $A$ that converges to $x$, (7) a function $f:\mathbb {R}\rightarrow \mathbb {R}$ is continuous at a point $x$ iff $f$ is sequentially continuous at $x$, (8) in $\mathbb {R}$, every unbounded set contains a countable, unbounded set, (9) the axiom of countable choice holds for subsets of $\mathbb {R}$.
LA - eng
KW - axiom of choice; axiom of countable choice; Lindelöf space; separable space; (sequential) continuity; (Dedekind-) finiteness; axiom of choice; axiom of countable choice; Lindelöf space; sequential continuity
UR - http://eudml.org/doc/248055
ER -