The product topology on $\{0,1\}^X$, where $X$ is uncountable

Question

I am trying to understand the topology on $\{0,1\}^X$, where $X$ is uncountable. The topology on $\{0,1\}$ is the discrete and I am using the product topology on $\{0,1\}^X$. My question is, who are the basic open sets? From my understanding of the definition of product topology, basic sets should either contain finite sets or cofinite sets (sets with finite complement). But from what written here nets and sequential spaces

in page 4 example 3, it seems like open basic sets must contain finite sets. Mabe I should ask, Who are the basic sets in this topology?

Thank you! Shir

Answer 1

As always with the product topology, a basis element will look like $\{0\}$ or $\{1\}$ (i.e., a typical open set for the topology of $\{0,1\}$) in finitely many components, and the whole set in all the remaining components.

Answer 2

I'll quote the beginning of the example from the document you have linked to.

Example 3: Let $X$ be an uncountable set and let $\{0,1\}$ have the discrete topology. Consider $\mathcal P(X) = \{0,1\}^X$ with the product topology. Let $\mathcal A\subseteq \mathcal P(X)$ be the collection of all uncountable subsets of $X$. $\mathcal A$ is not open; indeed every basic open contains finite sets. However, we claim that $\mathcal A$ is sequentially open.

The important point is notice that the author identifies $\{0,1\}^X$ with $\mathcal P(X)$. Indeed, there is a very natural bijection between these two sets (and it occurs frequently in mathematics, which might be why it is mentioned in the text without any details).

For every subset $S\subseteq X$ you have the corresponding function $\chi_S \in \{0,1\}^X$ which maps elements of $S$ to $1$ and other elements to $0$ $$\chi_S(x)= \begin{cases} 1 & x\in S, \\ 0 & \text{otherwise}. \end{cases} $$ The assignment $S\mapsto \chi_S$ is a bijection between $\mathcal P(X)$ and $\{0,1\}^X$.

Now if we look at the space $\{0,1\}^X$ then the basic open sets are obtained in this way: Choose finitely many coordinates $k_1,\dots,k_n$ and choose values $v_1,\dots,v_n\in\{0,1\}$ which you want to prescribe for these coordinates. Then you get a basic neighborhood consisting of all functions with these prescribed values, i.e. $f(k_j)=v_j$.

What happens if we transfer this topology (using the bijection we mentioned above) to the set $\mathcal P(X)$? Well, for every basic neighborhood we have fixed some finite set $F$ of indices where the functions have to attain the value $1$. If this function is of the form $\chi_S$, this is the same as saying $F\subseteq S$. We also prescribed some finite sets $G$ such that for $x\in G$ we want $\chi_S(x)=0$. This is the same as requiring $S\cap G=\emptyset$.

So the basic open sets in $\mathcal P(X)$ are like this: For any finite sets $F,G\subseteq X$ you get a basic set $$\mathcal O_{F,G}=\{S\subseteq X; F\subseteq S, G\cap S=\emptyset\}.$$ You can notice, that for any choice of $F$, $G$ such that $F\cap G=\emptyset$, this basic set contains at least one finite set. (For $F\cap G\ne\emptyset$, the set $\mathcal O_{F,G}$ described here is empty.)

Answer 3

It will be easier, perhaps, to think about it as a topology on the power set of $X$. The basic open sets are the sets of the form: $$U_{A,B}=\{Y\subseteq X\mid A\subseteq Y\land B\cap Y=\varnothing\}$$ Where $A,B$ are finite subsets of $X$.

Answer 4

You may start by defining a sub-basic open set by fixing an $i\in X$ and an open $U\subset\{0, 1\}$ (or $U$ can be taken from a topology basis of $\{0, 1\}$, or from a sub-basis; practically, $U = \{0\}$ or $U = \{1\}$), and setting

$$ O_{i, U} = \{ f\in\{0, 1\}^X\mid f(i)\in U\}. $$

A basic open will be the intersection of finitely many sub-basic opens. It has no importance if $X$ is countable, or uncountable, or finite non-empty.

Informally, to get a basic open in $\{0, 1\}^X$ (recall that elements of $\{0, 1\}^X$ are functions $X \to \{0, 1\}$) you fix a finite set of elements of $X$ and fix the function values on those elements; all the functions that match will form your open set.