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Thread: Path connected

  1. #1
    MHF Contributor
    Mar 2010

    Path connected

    If X and Y are path connected, show that XxY is also path connected.

    For a fixed x_0\in X and y_0\in Y
    Let A be the set of all the points that join x_0\times y_0 by a path in U\times V.

    Since X and Y are both path connected, there exist \epsilon >0 such that B(x,\epsilon) \ \text{and} \ B(y,\epsilon), where x is in \pi_1(A) and y is in \pi_2(A), contained in open sets U\subset X and V\subset Y, respectively.

    Let x_1\in B(x,\epsilon) then a path can be connected from x_1 to x to x_0.

    Similarly for y

    How can now show XxY is path connected?

    How can I translate this to the product topology?

    Can I say:

    x_1\times y_1\in B(x\times y,\epsilon)\subset A\subset U\times V. Therefore the exist a path from x_1\times y_1 to x\times y to x_0\times y_0????
    Last edited by dwsmith; Oct 17th 2011 at 03:23 PM.
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  2. #2
    Junior Member
    Sep 2011

    Re: Path connected

    Suppose X and Y are path connected. By definition, for any x_1,x_2\in X there is a continuous function f:[0,1]\rightarrow X with f(0)=x_1 and f(1)=x_2

    Now let (x_0,y_0) and (x_1,y_1) be two points in X\times Y and let f be the path in X from x_0 to x_1 and g be the path in Y from y_0 to y_1. Define h:[0,1]\rightarrow X\times Y by h(s) = (f(s),g(s)). Then h(0)=(x_0,y_0) and h(1)=(x_1,y_1).

    You should know a theorem that tells you h is continuous since it maps into a product space and each coordinate is given by a continuous function(Thm 19.6 in Munkres)
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