Cardinality of closed sets of reals

Question

set A is closed if for all sequences $\{a_n\}_{n=1}^{\infty}$ such that for each $n$ , $a_n \in A$ then if $\lim a_n$ exists then $\lim a_n \in A$

So what is the cardinality of closed sets of reals ?!!

Also i want to be sure $A^B$ is the carnality of all function $f : A \to B$ ?!

The meat of this question is answered in https://math.stackexchange.com/questions/740095 . — Chris Culter, May 11 '18 at 17:41
It is not quite clear from your phrasing whether you are asking what the sizes of individual closed sets can be, or what the size of the set of all closed sets is. — Andrés E. Caicedo, May 11 '18 at 19:18
You should avoid using both ! and ? at the same time, because in English it comes across as rude — Jack M, May 11 '18 at 20:48

mathcounterexamples.net · Accepted Answer · 2018-05-11T19:46:53.317

3

Let $C$ be the set of closed subsets of the reals. The map $\varphi : c \mapsto \mathbb R \setminus c$ is a bijection from $C$ onto the set $O$ of open real subsets of $\mathbb R$. Therefore the cardinality of $C$ is equal to the one of $O$.

Now, it is known that an open subset of the reals is a countable union of disjointed intervals. As the cardinality of the set of real intervals is the continuum, this is also the case of $O$ (the cardinality of $\mathbb R^{\mathbb N}$ is the one of $\mathbb R$). Finally we can conclude that the cardinality of $C$ is equal to the one of $\mathbb R$.

edited May 11 '18 at 19:46

answered May 11 '18 at 19:11

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can we show a bijection between the set $O$ and the set $R \times R$, by given a function $\phi ( (a,b)) = <a,b>$ ?! – Ahmad May 11 '18 at 19:38
@Ahmad, a subset of $\Bbb R\times\Bbb R$. What happens for $b < a$? – Martín-Blas Pérez Pinilla May 11 '18 at 20:01

Cardinality of closed sets of reals

1 Answers1