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Unit square

Is it possible to partition a unit square [0, 1] X [0, 1] into two disjoint
connected subsets A and B such that A and B contain opposing corners? I.e.,
such that A contains (0, 0) and (1, 1), and B contains (1, 0) and (0, 1)?

*----0
| |
| |
0----*

Evidently, A and B couldn't be path-connected because a path running from
(0, 0) to (1, 1) would intersect a path running from (1, 0) to (0, 1). So
what about connected, but not path-connected, subsets? (The topology on the
square is simply assumed to be the topology it inherits as a subspace of
euclidean R^2.)

Solution Summary

This solution is comprised of a detailed explanation to answer what about connected, but not path-connected, subsets?

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