Hi I am having problems with this:
Let $X$ be a countable discrete topological space $\{0,1 \}$, I need to find a dense countable set in $X$, but don´t know which se could be work, thanks for the help.
Hi I am having problems with this:
Let $X$ be a countable discrete topological space $\{0,1 \}$, I need to find a dense countable set in $X$, but don´t know which se could be work, thanks for the help.
HINT: Let $D=\{0,1\}$ with the discrete topology, for each $k\in\Bbb N$ let $D_k$ be a copy of $D$, and let $X=\prod_{k\in\Bbb N}D_k$.
For each $x\in X$ let $s(x)=\{k\in\Bbb N:x_k=1\}$, and let $A=\{x\in X:s(x)\text{ is finite}\}$; show that $A$ is a countable dense subset of $X$.
Added: Let $\Sigma$ be the set of finite sequences of $0$’s and $1$’s. For each $\sigma=\langle b_0,\dots,b_n\rangle\in\Sigma$ let $B(\sigma)=\{x\in X:x_k=b_k\text{ for }k=0,\dots,n\}\;,$ and let $\mathscr{B}=\{B(\sigma):\sigma\in\Sigma\}$; $\mathscr{B}$ is a base for the product topology on $X$, so to show that $A$ is dense in $X$, we need only show that $B\cap A\ne\varnothing$ for each $B\in\mathscr{B}$.
Fix $\sigma=\langle b_0,\dots,b_n\rangle\in\Sigma$, and let $F=\{k\le n:b_k=1\}$. Define a point $x\in X$ as follows: for each $k\in\Bbb N$,
$x_k=\begin{cases} 1,&\text{if }k\in F\\ 0,&\text{otherwise}\;; \end{cases}$
then $s(x)=F$, so $x\in A\cap B(\sigma)$, as desired.