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Let X be path-connected and let b∈X . Show that every path in X is homotopic...

Let X be path-connected and let b∈X . Show that every path in X is homotopic with endpoints fixed to a path passing through b

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Answer #1

Let be a path with fixed end point say, x and y. Then we need to find a path with the same end point passing through b, which is homotopic to . Since X is path connected there is a path from x to b.

Define , by , , and

Then note that is a path passing through b. Also note that and are homotopic, as .

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