Definition. Let X be a topological space and suppose xo is a point in X. A map exo : [0, 1] → X that sends every point of [0, 1] to the single point x, is called a constant path. Theorem 12.7. Let a be a path from x, to x1. Then ex, · a ~ a and a · ex, ~ a. To prove the above theorem, you will want to construct homotopies that demon- strate the equivalences. Recalling the definition of the product, notice that ex,' a is a path that sits still at x, while s runs from 0 to 1/2 and then moves along a (twice as fast as usual) while s runs from 1/2 to 1. To write down a homotopy, it may help to think about what the intermediate paths might look like between this and a. You'll have many choices, so you might as well choose a homotopy that is easy to write down. If we think of a as tracing out a path, then tracing out that same image in reverse yields a natural inverse.
Definition. Let X be a topological space and suppose xo is a point in X. A map exo : [0, 1] → X that sends every point of [0, 1] to the single point x, is called a constant path. Theorem 12.7. Let a be a path from x, to x1. Then ex, · a ~ a and a · ex, ~ a. To prove the above theorem, you will want to construct homotopies that demon- strate the equivalences. Recalling the definition of the product, notice that ex,' a is a path that sits still at x, while s runs from 0 to 1/2 and then moves along a (twice as fast as usual) while s runs from 1/2 to 1. To write down a homotopy, it may help to think about what the intermediate paths might look like between this and a. You'll have many choices, so you might as well choose a homotopy that is easy to write down. If we think of a as tracing out a path, then tracing out that same image in reverse yields a natural inverse.
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Could you explain how to do 12.7 with detailed explanation? Thank you!
![**Constant Path Definition:**
Let \( X \) be a topological space and suppose \( x_0 \) is a point in \( X \). A map \( e_{x_0} : [0,1] \rightarrow X \) that sends every point of \([0,1]\) to the single point \( x_0 \) is called a constant path.
**Theorem 12.7:**
Let \( \alpha \) be a path from \( x_0 \) to \( x_1 \). Then \( e_{x_0} \cdot \alpha \sim \alpha \) and \( \alpha \cdot e_{x_1} \sim \alpha \).
To prove the above theorem, you will want to construct homotopies that demonstrate the equivalences. Recalling the definition of the product, notice that \( e_{x_0} \cdot \alpha \) is a path that sits still at \( x_0 \) while \( s \) runs from 0 to 1/2 and then moves along \( \alpha \) (twice as fast as usual) while \( s \) runs from 1/2 to 1. To write down a homotopy, it may help to think about what the intermediate paths might look like between this and \( \alpha \). You’ll have many choices, so you might as well choose a homotopy that is easy to write down.
If we think of \( \alpha \) as tracing out a path, then tracing out that same image in reverse yields a natural inverse.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56be79ad-be6a-45f1-b897-58d23fd7e62d%2F5c66d965-c00b-49b0-8db2-0139ecc53e66%2Fmhbcj4j_processed.png&w=3840&q=75)
Transcribed Image Text:**Constant Path Definition:**
Let \( X \) be a topological space and suppose \( x_0 \) is a point in \( X \). A map \( e_{x_0} : [0,1] \rightarrow X \) that sends every point of \([0,1]\) to the single point \( x_0 \) is called a constant path.
**Theorem 12.7:**
Let \( \alpha \) be a path from \( x_0 \) to \( x_1 \). Then \( e_{x_0} \cdot \alpha \sim \alpha \) and \( \alpha \cdot e_{x_1} \sim \alpha \).
To prove the above theorem, you will want to construct homotopies that demonstrate the equivalences. Recalling the definition of the product, notice that \( e_{x_0} \cdot \alpha \) is a path that sits still at \( x_0 \) while \( s \) runs from 0 to 1/2 and then moves along \( \alpha \) (twice as fast as usual) while \( s \) runs from 1/2 to 1. To write down a homotopy, it may help to think about what the intermediate paths might look like between this and \( \alpha \). You’ll have many choices, so you might as well choose a homotopy that is easy to write down.
If we think of \( \alpha \) as tracing out a path, then tracing out that same image in reverse yields a natural inverse.
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