Let $X$ be Hausdorff space and $g: X \to X$ a homeomorphism.Let $G\cong \mathbb{Z}$ the group of homeomorphism of $\mathbb{R}\times X$ generated by $(t,x)\to (t+1,g(x))$.Prove that $\mathbb{R}\times X/G$ is Hausdorff.
I think that I can prove it without take $x,y\in \mathbb{R}\times X/G$ and construct $U_{x},V_{y}$ disjoint open neigborhoods of $x$ and $y$.
Try apply the following results that I already prove using that $\mathbb{R}$ is Hausdorff and $X \times \mathbb{R} $ should be Hausdorff:
Let $G$ be a group of homeomorphisms of a Hausdorff space $X$ and $\pi : X \to X/G$ the canonical map. Suppose there is an open subset $A \subset X$ such that:
- the quotient map $\pi : A \to X/G$ is onto;
- $\lbrace g \in G \mid g \cdot A\cap A =\emptyset \rbrace$ is finite.
Then $X/G$ is Hausdorff.
But I can´t find these $A$ for that reason I have some quiestions.
Question 1
Is possibly apply this result to this proof? In this case any Hint was welcome.
Question 2 In case of the answer of question $1$ were no. How I should start the proof I thought about it, but I don´t have idea of how I should construct the $U_x,V_x$, any idea or advice of how make the proof was useful, because I´m starting to learn this concepts and definitions.
Thanks a lot, for read me.