Range space and homotopy

Question

The main question is about range space and homotopy, but I also have another two small questions.

1. Are $f$ and $h$ from $\Bbb R \to \Bbb R^2-\{0\}$?
2. What are the range spaces of $f$ and $h$? $[0,1]$? How can we determine $f$ and $h$ are not homotopic from the range space? I think if we graph $f$ and $h$ separately, the images look the "same".
3. What's the meaning of those arrows on the graph of Figure 51.4? Why is the direction of $f$ different from $h$?

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Edit: What does the straight-line homotopy look like between $f$ and $h$ if we consider the space $\Bbb R^2$? The line segments which is parallel to the $y$-axis and bounded by the circle?

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Answer 1

First, note that the written example corresponds to Figure 51.4 and seems to have nothing to do with Figure 51.3.

1. $f$ and $h$ are paths in $X$, i.e. continuous functions from $[0,1]$ to $\mathbb{R}^2 - \{0\}$.
2. The range space of a path $\gamma$ in a topological space $X$ is $X$ -- in this case, since $X = \mathbb{R}^2 - \{0\}$ and $f$ and $g$ are paths in $X$, the range space of both $f$ and $g$ is $\mathbb{R}^2 - \{0\}$.
3. You can tell the direction of a path by looking at its parametrization. In this case, the $y$ coordinate of the path $f$ is positive for all $s \in [0,1]$, while the $y$ coordinate of $h$ is negative for all $s \in [0,1]$. Think about putting your pencil down at $f(0)$ (a point in $\mathbb{R}^2 - \{0\}$) and then "drawing" $f$ by advancing $s$ from 0 to 1. You'll end up with what's shown in the figure.

In general, determining which paths are homotopic in a particular topological space is a difficult question; your textbook likely has at least one chapter dedicated to the topic.

Answer 2

1. $f$, $g$ and $h$ are all maps $[0,1]\to \Bbb R\setminus\{0\}$.
2. Range space is another word for codomain. In that case, it is $\Bbb R\setminus\{0\}$. The domain, however, is $[0,1]$.
3. The arrows are here to remember the orientation of the domain $[0,1]$, "from $s=0$ to $s=1$". A homotopy has to respect this orientation and therefore you can "see" homotopies more easily if you see the orientation.