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If $\varphi:\mathbb T^{2}\rightarrow\mathbb C$ is a continuous function of two variables on the torus, then the range of $\varphi$ is always a closed curve?

Marcos Ferreira
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2 Answers2

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You need to specify what you mean by a curve. For instance in some context of analytic--or differential--geometry a curve is a map $$ f:I\longrightarrow \Bbb R^2 $$ where $I\subset\Bbb R$ is an interval (i.e. a curve is a function not a set)

Or else, a curve may be thought as a $1$-dimensional real variety, whereas in an algebraic context a curve is often a $1$-dimensional complex variety.

In any event, consider that the torus is compact, so the image of your map is going to be a compact subset of $\Bbb C$, hence closed and bounded.

As an example, embed the torus in $\Bbb R^3$ so that $\Bbb R^2$ is identified to a plane splitting it in half transversally.

Now let $\varphi$ the function on the torus that is the projection from $\Bbb R^3$ to $\Bbb R^2$. Is the image a curve in any sense?

Andrea Mori
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  • Hi Andrea Mori. Sorry! In fact, I am referring to a curve as a subset of $\mathbb C$. For instance, if $\gamma$ is a continuous complex-valued function on $\mathbb T$, then $\gamma$ is a closed curve (in the sense that the end point coincides with the start point). In this case, we can calculate the winding number of $\gamma$ about a point that is not in the range of $\gamma$. What I would like to know is if, in the case of continuous functions in bi-torus, we can have this same notion. Thank you. – Marcos Ferreira Jul 29 '19 at 13:48
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Taking the most obvious definitions for the words, the answer is no. A torus is a two-dimensional object. Generally, the image will have two dimensions, so it will not be a curve.

Jiri Lebl
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    To give an explicit example, lay a doughnut flat on the table, and flatten it. The image is an annulus. – JonathanZ Jul 31 '19 at 18:11