TITLE: topology related to Binary space
QUESTION [5 upvotes]: I ran upon this topology based on binary space, perhaps using obscure terminology, but I am curious what it is and its properties. 
Let binary space be the set of strings of $0,1$'s, and let $S$ be the set of all functions that map the binary space to a set of two elements, $\{0,1\}$. It's like it decides a true or false for every binary string. For the topology $\mathcal{T}$, let a basic set be $U_{V,f}$, where any $g$ in this set must satisfy $g(x)=f(x)$ for all $x$ in $V$, and $V$ is a finite subset of $B$. 
Is this space $(S,\mathcal{T})$ Hausdorff? Compact? Any related material is welcome.
Thanks

REPLY [6 votes]: The set of finite binary sequences $B$ is denoted $\{0,1\}^{<\omega}$. Your set $S$ is the set of functions $f:\{0,1\}^{<\omega}\to \{0,1\}$. That is, $$S=\{0,1\}^{\{0,1\}^{<\omega}}.$$
You defined a basic open set around $f\in S$ to be the set of all $g\in S$ which agree with $f$ on a given finite subset of ${\{0,1\}^{<\omega}}$. This is exactly how basic open sets are defined in the product topology. As the product of compact Hausdorff spaces (in your case $\{0,1\}$) is compact Hausdorff in the product topology, $S$ is compact Hausdorff.  Do not get hung up on the index set $\{0,1\}^{<\omega}$. You could replace it with any other countably infinite set (such as $\omega$) and get the same space.