On the definition of Jordan curves

Question

I read that the definition of Jordan curve is that it is homeomorphic to $S^1$. Is this equivalent to say that the curve is closable, continuous and non-self-intersecting? I'm not sure if closable is the correct term but I mean that if we have a curve $f$ from $[a,b]\to X$ for some space $X$ then curve is closable if $f(a)=f(b)$. And by non-self-intersecting I mean that we only connect endpoints of the line segment together but the curve does not intersect itself in any other points. Hopefully you understand.

Answer 1

To put it mathematically: A continuous map $ f\colon[a,b]\to\boldsymbol{R}^2$ is called a Jordan curve iff $f(a)=f(b)$ and $f$ is injective on $[a,b[$.

Answer 2

So we have two definition.

A Jordan curve is a subpace of the plane homeomorphic to the circle $S^1$.

and

A simple closed curve is a continuous map $\gamma\colon [a,b]\to\mathbb{R}^2$ such that $\gamma(a)=\gamma(b)$ and for all $c\in (a,b)$, if there exists a $d\in[a,b]$ such that $\gamma(c)=\gamma(d)$ then $c=d$.

By the universal property of quotient maps, given a surjective quotient map $q\colon [a,b]\to S^1$ which is injective on $(a,b)$, for every simple closed curve $\gamma$, there exists an injective map $\tilde{\gamma}\colon S^1\to\mathbb{R}^2$ such that $\tilde{\gamma}\circ q=\gamma$.

Note that $q$ is surjective and also $\tilde{\gamma}\circ q$ is surjective onto the image of $\gamma$. In particular, $\tilde{\gamma}$ is surjective onto the image of $\gamma$ and by construction it is injective, hence $\tilde{\gamma}$ is a continuous bijection from the compact space $S^1$ to the Hausdorff space $\mbox{Im } \gamma$ (Hausdorff because it is a subspace of the Hasudorff space $\mathbb{R}^2$). It follows that $\tilde{\gamma}$ is a homeomorphism from the circle to the image of $\gamma$ and so the image of every simple closed curve is homeomorphic to the circle.

In short, they are equivalent concepts except one is given in terms of a map, and the other is given in terms of a subspace (which can be parametrised by an identical map).