When is $\bold N$ Lindelöf?
Herrlich, Horst ; Strecker, George E.
Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997), p. 553-556 / Harvested from Czech Digital Mathematics Library
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Theorem. In ZF (i.e., Zermelo-Fraenkel set theory without the axiom of choice) the following conditions are equivalent: (1) $\Bbb N$ is a Lindelöf space, (2) $\Bbb Q$ is a Lindelöf space, (3) $\Bbb R$ is a Lindelöf space, (4) every topological space with a countable base is a Lindelöf space, (5) every subspace of $\Bbb R$ is separable, (6) in $\Bbb R$, 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 $f:\Bbb R\rightarrow \Bbb R$ is continuous at a point $x$ iff $f$ is sequentially continuous at $x$, (8) in $\Bbb R$, every unbounded set contains a countable, unbounded set, (9) the axiom of countable choice holds for subsets of $\Bbb R$.

Publié le : 1997-01-01
Classification:  03E25,  04A25,  26A03,  26A15,  54A35,  54D20 
@article{118952,
     author = {Horst Herrlich and George E. Strecker},
     title = {When is $\bold N$ Lindel\"of?},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {38},
     year = {1997},
     pages = {553-556},
     zbl = {0938.54008},
     mrnumber = {1485075},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118952}
}

Herrlich, Horst; Strecker, George E. When is $\bold N$ Lindelöf?. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 553-556. http://gdmltest.u-ga.fr/item/118952/

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