I am a beginner in algebraic topology. I am solving the following exercise from my book:
Problem: Consider space $X:=\mathbb{R^2} \setminus \{(0,0)\}$. Define space $\tilde{X}$ with quotient topology $\tilde{X} \to X$ as follows:
$\tilde{X}=\{[\gamma]: \gamma$ is a path in $X=\mathbb{R^2}\setminus\{(0,0)\}$ starting from a point (fixed)$x_0\in X$ $\}$.
and $[\gamma]=\{\gamma': \gamma$ is path homotopic to $ \gamma'\} $ , for simply connected open sets $U\subset X$ basis for the topology on $\tilde{X} $ are sets as follows:
$U_{[\gamma]}=\{[\gamma \cdot \eta]: \eta$ is a path in $U$ and $\gamma (1)=\eta (0)\}$
I want to show $X$ and $\tilde{X}$ are not homeomorphic.
My work:
My plan of attack is to find some invariant topological property that holds in one space but does not in other.
Any hint or a reference will be a great help!!