Is semi-circle a Jordan curve? The equation of the semi circle is given by; $z(t)=z_0+r e^{it}, t \in  [0,\pi]$ , where $r > 0$ is radius and $z_0$ is the center of the semi circle.
Clearly, a semi circle is not a closed curve although it is simple. So, it should not be a Jordan curve. But our professor told us that it is a Jordon curve. Which one is correct?
Also, she gave the following definition of a Jordan curve:

$ \Gamma$ is said to be a Jordon Curve or a simple curve if it has no multiple point except when it is closed in which case it has exactly one double point i.e., for any two points $t_{1}, t_{2} \in[\alpha, \beta]$ s.t. $t_{1}<t_{2}, z\left(t_{1}\right) \neq z\left(t_{2}\right)$ except possibly ${t}_{1}=\alpha, t_{2}=\beta$.

I think, she considered in the definition of Jordan curve that it may not be necessarily closed and that's why she told that a semi circle is a Jordan curve. But, on google and in every book I referred to, it is told that a Jordan curve is a closed and simple curve.
I am really confused as to which definition is correct. Thankyou for any help.
 A: A definition cannot be wrong because everybody is free to define anything. But the definition of "Jordan curve" in your question in highly non-standard. Usually one understands it as a closed curve $\Gamma : [\alpha,\beta] \to \mathbb C$ without multiple points (i.e. $\Gamma(\alpha) = \Gamma(\beta)$ and $\Gamma(t_1) \ne \Gamma(t_2)$ for $\lvert t_1 - t_2 \rvert < \beta - \alpha$). In your definition the closedness condition $\Gamma(\alpha) = \Gamma(\beta)$ is omitted, thus Jordan curves may be closed or non-closed (this means that $\Gamma$ is injective). Non-closed Jordan curves are usually called Jordan arcs.
Which lesson can be learnt from this?
Mathematical notation is not as standardized as one might expect. Even in textbooks one must be aware that authors might use the same denotation for slightly different things. Therefore, if something appears to be strange, carefully read the definition and compare it with other definitions you have been reading elsewhere. If you find a discrepancy, do not believe that something is wrong, but accept it as it is. Yes, it may be a "minority view", but it is nevertheless legitimate in its context. Learn to be flexible and understand the concept behind denotation.
