Problem
Let $X$ be a path-connected topological space and suppose that given two points $a,b$ the image of every path connecting $a$ to $b$ intersects some subset $S$. Is $S$ connected?
Context / Digression
In my Analysis class there was a problem that looked like the following.
Let $D \subset \mathbb{R}^2$ be the closed unit disk and $a = (-1,0), b = (1,0)$ and suppose $f \colon D \to \mathbb{R}$ is a continuous function where $f(a) = -10$ and $f(b) = 10$.
The two parts was that we had to show that $f$ was equal to 0 somewhere on $D$ then we were challenged to find out how many zeros there were.
We know there is a zero because the image will be a closed (because $D$ is compact) interval (because $D$ is connected) where the containing $0$ (because it contains both $-10$ and $10$). We know there are uncountably many zeros as there are uncountably many paths from $a$ to $b$ (there is at least one path intersecting each point of the $y$-axis) and each path must contain a zero as the map $[0,1] \to \mathbb{R}$ will similarly be a closed interval containing $0$.
In class I felt like there was more to say about the set of points where $f$ was $0$. I actually felt like the zero set would contain a subset $S$ such that $S$ is connected and every function would have to pass through $S$. I felt this way intuitively because if there was a path connecting $a$ to $b$ without intersecting the set then there would be a disconnection $U,V$ where $U$ is the set of points with $y$ values greater than the $y$ value of the path and $V$ contains all of the lower points. So this construction does not work because every such path contains a zero. But this is not rigorous because it can be the case that such a disconnection exists without using my $U$ and $V$.
There are more questions this problem too. For example, would the space $X - S$ be disconnected? Do we need the path-connected hypothesis to say $S$ is connected (if the answer to the original question is yes). But that's getting ahead of myself. Any progress on this problem is greatly appreciated.
Let me know if anything is unclear.
The answer is no. Let $X$ be the unit circle in $\Bbb R^2$, $a=(1,0)$, $b=(-1,0)$ and $S=\{(0,1),(0,-1)\}$.
Note also that $X\setminus S$ can be connected, for example for $X$ the Warsaw Circle and $S$ a point on the nice part of the arc.