For questions relating to path-connected topological spaces, that is, spaces where any two points can be connected by a path.

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Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix. Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
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Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?

An anonymous question from the 20-questions seminar: Can you explicitly write $\mathbb{R}^2$ as a disjoint union of two totally path disconnected sets?