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Path connected subsets

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Give proofs or counter-example for the following statements:
i)If X and Y are path connected subsets of Z and X/Y ( X intersection Y)

is non empty then X/Y is path connected.

ii) If X and Y are path connected subsets of Z the X/Y(union) is

path-connected.

iii)If X and Y are path connected subsets of Z and X/Y ( X intersection Y)

is non empty then X/Y(union) is path connected.

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https://brainmass.com/math/algebraic-geometry/path-connected-subsets-87161

Solution Preview

i) False. Take as X and Y as the two halves of a circle, that intersect
in 2 distinct points.

ii) False. Take X = {0}, Y = {1} in R.

iii) ...

Solution Summary

This solution is comprised of a detailed explanation to give proofs or counter-example for the following statements.

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