# Show that the existence of a simple closed curve is a topological property.

A simple closed curve on a topological space $$X$$ is a continuous map $$γ : [0, 1] → X$$ such that $$γ(0) = γ(1)$$ and $$γ|_{[0,1)}$$ is one-to-one.

Show that the existence of a simple closed curve is a topological property.

I know that a topological property is defined to be a property that is preserved under a homeomorphism. Examples are connectedness and compactness.

So how do I show connectedness and compactness in here? Are there other properties?

Let $$\psi:X\to Y$$ be a homeomorphism and $$\gamma: [0,1]\to X$$ be a simple closed curve in $$X$$. Then, $$\psi\circ \gamma:[0,1]\to Y$$ is a simple closed curve in $$Y$$.
Note that $$\psi\circ \gamma$$ is continuous as the composition of two continuous functions is continuous.
Next, $$\psi\circ \gamma(0)=\psi\big(\gamma(0)\big)=\psi\big(\gamma(1)\big)=\psi\circ \gamma(1)$$.
Finally, $$\gamma\big|_{[0,1)}$$ is injective and $$\psi$$ is injective imply $$\psi\circ \gamma\big|_{[0,1)}=\big(\psi\circ \gamma\big)\big|_{[0,1)}$$ is injective.
Similarly, for any simple closed curve $$\delta:[0,1]\to Y$$ we can say $$\psi^{-1}\circ \delta:[0,1]\to X$$ is a simple closed curve in $$X$$.
What you are asked to show is that if $$X$$ is homeomorphic to $$Y$$ and there is a simple closed curve $$\gamma$$ in $$X$$ then there is simple closed curve in $$Y$$ too. Let $$\phi:X \to Y$$ be a homeom orphism. Define $$\gamma '(t)=\phi (\gamma (t))$$ for all $$t \in [0,1]$$. Verify that $$\gamma '$$ is a simple closed curve in $$Y$$.