Vtyjkyuk

For an alphabet
$A=lbrace 1,2,3,...,k-1rbrace$ the set $A^mathbbN=Atimes Atimes Atimescdots$ consisting of all (one-sided) sequences of
elements of $A$. The shift map $sigma:A^mathbbNrightarrow A^mathbbN$ is defined by $sigma(x_0x_1x_2cdots)=x_1x_2cdots$, which is continuous on this
space. We called $(A^mathbbN,sigma)$ the full $A$- shift space. We defined a metric $d$ on $A^mathbbN$ by $$ begineqnarray
d(s,t)=left{
beginarrayll
0, & s = t,nonumber
\
2^-j, & sneq t.
endarrayright.
endeqnarray$$ where $j in N$ is the minimal number such that $s_j neq t_j$. The space $(A^mathbbN,d)$ is a topological space. So
the basic open ball $W$ is any subset of the full $A-$ shift of the form $$ W=X_W lbrace xin Sigma_k:x_0x_1x_2cdots x_k-1=wrbrace$$, where $w$ is an allowed block of length $k$. My question is how can we define a supra topology on $A^mathbbN$ ?. Where the supra topology $tau^*$ is a collection of subsets of the $P(X)$ such that $X$ and $emptyset$ are in $tau^*$ and $tau^*$ is closed under arbitrary union.

The topology is defined by the metric.

As the shift operator is not needed to define the metric, calling this the shift space is a misnomer. This space is an infinite string space.

A base for this space is all sets of strings with a common prefix. Notice that the space you defined does not allow for finite strings.

You ask for a supra topology with such scant requirements, that any topology whatever is a supra topology.