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Math Help - Two pathwise connected problem TEST TOMORROW, PLEASE HELP!!!

  1. #1
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    Two pathwise connected problem TEST TOMORROW, PLEASE HELP!!!

    1) Let A, B be subsets of R^n that are pathwise connected with A interset B not empty. Prove A U B is pathwise connected.

    2) Prove S = {(x,y) in R^2 : either x or y is rational number} is pathwise connected.

    I have been studying for two weeks, and will continue to study tonight, but I simply cannot study some problems that I do not have the answers for.

    Thank you!
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  2. #2
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    #1 is easy. If x & y are in AUB then you have two cases.
    If both x & y are in A or in B then there is an arc from x to y. WHY?
    If x is in A and y is in B then there is a z in A intersect B. There is an arc from x to z and there is an arc from z to y. The union of those connects x to y with a path.

    For #2, I would need to know what theorems about paths(arcs) you have.
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  3. #3
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    So far, our theormes on pathwise connectedness are:

    1. If A is pathwise connected, and f:A->R^n is continuous, then f(A) is pathwise connected.
    2. If A is convex, then A is pathwise connected.
    3. A is pathwise connected iff A is an interval.

    I think that is all so far.
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