Suppose I have a topological space $S$ and a continuous real-valued function $f:S \to \mathbb R$. I can define sets like: \begin{align} A &= \{x \in S : f(x) = 0 \} \\ B &= \{x \in S : f(x) \le 0 \} \\ C &= \{x \in S : f(x) < 0 \} \\ \end{align} What can we say about the topological relationships between $A, B, C$?
For example, is it true that $A$ is the boundary of $B$, or that $C$ is the interior of $B$. If those statements are not true, and we want them to be true, what extra conditions do we have to impose on $f$ or on the topology of $S$. I’d be happy to assume that $S$ is a metric space, or even a normed space, for example.
Context: these sorts of sets are used to model shapes in computer systems, so I’m interested in their topology.
You can't really say much without knowing the specifics of $f$. For example, even in a nice space like $S = \Bbb R$, consider the continuous function $f : \Bbb R \to \Bbb R$ $$ f(x) := \begin{cases} -1 &x \leq -1 \\ x &-1 < x < 0 \\ 0 &x \geq 0 \end{cases} $$ Then \begin{align} A &= \{x \in \Bbb R : f(x) = 0 \} = [0, +\infty) \\ B &= \{x \in \Bbb R : f(x) \le 0 \} = \Bbb R \\ C &= \{x \in \Bbb R : f(x) < 0 \} = (-\infty, 0) \\ \end{align}
So certainly $B$ is not the closure of $C$ and $A$ is not the boundary of $C$.
However, there is a condition that you can put of $f : S \to \Bbb R$ to ensure that $B = \overline{C}$ (for arbitrary topological space $S$). If $f$ is an open map in addition to being continuous, then $$ \overline{f^{-1}(U)} = f^{-1}(\overline{U}) $$ is true. So $$ \overline{C} = \overline{f^{-1}((-\infty, 0))} = f^{-1}(\overline{(-\infty, 0)}) = f^{-1}((-\infty, 0]) = B $$