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


*

*Are $f$ and $h$ from $\Bbb R \to \Bbb R^2-\{0\}$?

*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".

*What's the meaning of those arrows on the graph of Figure 51.4? Why is the direction of $f$ different from $h$?



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? 


 A: *

*$f$, $g$ and $h$ are all maps $[0,1]\to \Bbb R\setminus\{0\}$.

*Range space is another word for codomain. In that case, it is $\Bbb R\setminus\{0\}$. The domain, however, is $[0,1]$.

*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.

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


*

*$f$ and $h$ are paths in $X$, i.e. continuous functions from $[0,1]$ to $\mathbb{R}^2 - \{0\}$.

*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 $\mathbb{R}^2 - \{0\}$.

*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.
