under consideration is: $\mathbb{1}_{[0,1)}:\mathbb{R}\to\{0,1\}$ $$\mathbb{1}_{[0,1)}(x)= \begin{cases} 1,& 0\leq x<1\\ 0,& \text{otherwise} \end{cases}$$ My first question is that I don't know what the 'default' topology on $\{0,1\}$ is? Is it the discrete topology, or the Sierpiński topology?
My second question is most clearly formulated in general terms first:
If $f:X\to Y$ a map between topological space, then what topology is implicitly used on $A$ in the restriction $f|A:A\to Y$?
Specifically in this example we consider $\mathbb{1}_{[0,1)}\lvert{[0,1]}:[0,1]\to\{0,1\}$. Then what is the topology defined on $[0,1]$? I suspect it is the subspace topology, so that $[0,0.1)$ for example is open.
The default topology on $\{0,1\}$ is the subspace topology induced from $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$, i.e. the discrete topology. If a different topology is considered, that is (usually, at least) explicitly mentioned. Typically, characteristic functions are considered in contexts where other real- or complex-valued functions occur too, and one adds or multiplies these functions, so one considers characteristic functions as real- or complex-valued, only the range happens to be contained in $\{0,1\}$.
For the second question, when considering the restriction of a map $f\colon X \to Y$ between two topological spaces to $A\subset X$, the topology on $A$ one considers is, unless explicitly stated otherwise, the subspace topology on $A$ induced by $X$.