Continuous functions on the torus.

Question

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?

Answer 1

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?

Answer 2

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.