# Proof that the space of infinite 01-sequences (Cantor-space) is totally disconnected

I want to proof that the space $\{0,1\}^{\mathbb N}$ of infinite binary sequences with the product topology is totally disconnected. I know that this space has a basis consisting of clopen sets and is $T_2$, so it follows already that it is totally disconnected. But I tried to proof it directly using the definition.

Def: A topological space $X$ is totally disconnected iff only the singletons are connected, i.e. if for every set $Y$ with more than two elements there exists two nonempty separated sets $X_1, X_2$ such that $Y = X_1 \cup X_2$.

Proof: Let $Y \ne X$ be a set with more than one element. Then if $Y$ is finite, select some $y \in Y$, and because for finite sets $M$ it holds that $cl(M) = M$ it follows that $Y = \{y\} \cup Y\setminus\{y\}$ is a separation. Otherwise $Y$ is infinite if we can select a point $y$ which is not a limit point of $Y$ and we can write again $Y = \{y\} \cup Y\setminus\{y\}$.

I am not sure if it is always possible to select a point which is not a limit point of $Y$, but I had sets like $Y = \{ ab^{\mathbb N}, aab^{\mathbb N},aaab^{\mathbb N},\ldots \}$ in mind, which has limit point $a^{\mathbb N}$. Is there a way to proceed along the lines of this proof and construct for every $Y$ such a partition into separated sets?

• "I am not sure if it is always possible to select a point which is not a limit point of $Y$" It isn't. The space has perfect proper subsets. – Daniel Fischer Sep 20 '13 at 13:35
• I would argue: Let $x \in X$, and $C$ the connected component of $X$. Since the projection $\pi_k$ to the $k$-th coordinate is continuous, we have $\pi_k(C) \subset \pi_k(\{x\})$. Since that holds for all $k$, $C \subset \prod \pi_k^{-1}(x_k) = \{x\}$. – Daniel Fischer Sep 20 '13 at 13:38

In $\{0,1\}^\Bbb{N}$ let $A$ be a subset with at least two elements $(a_n)_n$ and $(b_n)_n.$ There is an index $k$ where $a_k\ne b_k.$ But then $p_k^{-1}(0)$ and $p_k^{-1}(1)$ are disjoint open sets ($p_k$ is the projection onto the $k$-th coordinate), one containing $(a_n)_n$ and the other one containing $(b_n)_n$, and both cover $A.$