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Path connected problems

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? Show that, for , the sphere is path connected.
? Show that if f:X->Y is a continuous map between topological spaces and X is path connected, then the image f(Y) is also path connected.
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This solution is comprised of a detailed explanation to
? Show that, for , the sphere is path connected.
? Show that if f:X->Y is a continuous map between topological spaces and X is path connected, then the image f(Y) is also path connected.

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1 a. Three cities are at the vertices of and equilateral triangle of unit length. Flying Executive Airlines needs to supply connecting services between these three cities. What is the minimum length of the two routes needed to supply the connecting service?

1 b. Now suppose Flying Executive Airlines adds a hub at the "center" of the equilateral triangle. Show that the length of the routes needed to connect the three cities has decreased by 13%. (Note: It has been shown that no matter how many "hubs" you add and no matter how many points must be connected, you can never save more than 13% of the total distance needed to "span" all the original points by adding hubs.)

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