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Math Help - Digital line topology

  1. #1
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    Digital line topology

    Prove that the digital line topology is path connected.

    (the D.L.T is the topology has basis elements {n} and {m-1,m, m+1}, for n odd, m even)
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Prove that the digital line topology (on Z ?) is path connected.

    (the D.L.T is the topology has basis elements {n} and {m-1,m, m+1}, for n odd, m even)
    Definition. A space X is path connected if every pair of points of X can be joined by a path in X.

    Lemma 1. Let \{A_n\}_{n=1}^{\infty} be a sequence of path connected subsets of a space X such that for each integer n>=1, A_n has at least one point in common with one of the preceding sets A_1, ...,A_{n-1}. Then, \bigcup_{n=1}^{\infty}A_n is path connected.

    Let T be a digital topology on \mathbb{Z}. Without loss of generality, let x, y be arbitrary pair of points (x \leq y ) in \mathbb{Z}.
    Choose A_1 be {x} if x is odd, or {x-1, x, x+1} if x is even. Since every basis element of (\mathbb{Z}, T) is path connected (verify this), if y belongs to A_1, then we are done. Otherwise choose A_2 be {x, x+1, x+2} if x is odd, or {x+1} if x is even. This process continues until we find A_{y-x+1} that contains y. Since each A_n and A_{n+1}, n=1, 2, ... ,y-x, has an intersection and is path connected, we conclude that (\mathbb{Z}, T) is path connected by lemma 1.
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