$A, B$ are bounded, connected subsets of $\mathbb{R}^2$ with non-empty interiors. Consider their shared boundary, i.e. the set:
$$S:= bd\;A \cap bd\;B$$
Define:
$$ S_1 = \{(x,x):x \in [0.2,0.3]\}\\ S_2 = \{(x,x):x \in [0.4,0.5]\}. $$
It is given that (1) $S_1$ and $S_2$ are subsets of $S$, (2) the boundary points of $S_1$ and $S_2$ are boundary points of $S$ and (3) $\{(x,x):x \in (0.3,0.4)\}\cap S = \emptyset$.
My question is: Can we say that there exist two paths -- say $\gamma_1$ and $\gamma_2$ -- between the "top" endpoint of $S_1,(0.3,0.3),$ and the "bottom" endpoint of $S_2,(0.4,0.4),$ such that $\gamma_1 \subseteq bd\;A$, $\gamma_2 \subseteq bd\;B$ and $\gamma_1 \cap \gamma_2 = \{(0.3,0.3),(0.4,0.4)\}$?
I have basic knowledge of metric spaces and topology but I'm new to the notions of connectedness and paths. A picture is attached to capture the way I've visualized this so far. Any help is most appreciated.
Not always - it depends on what $A$ and $B$ are. You've drawn it in a way that makes $\gamma_1$ and $\gamma_2$ exist. But if you take $A = B$ to be a set that looks like the following:
Here, $S = \text{bd}(A) = \text{bd}(B)$ is just the boundary, and we can see that all the initial conditions are met. There is no way to draw a path from $(0.3,0.3)$ to $(0.4,0.4)$ that remains in $S$ since $S$ is not connected.