Let $X$ be the quotient space $[0,1]/\sim$ where $\sim$ is the equivalence relation given by $0\sim 1\sim 1/2\sim \cdots\sim 1/n \sim \cdots$.
Let $H$ (the Hawaiian Earring, aka the Infinite Earring) be the subspace of $\mathbb{R}^2$ consisting of the union of circles of radius $1/n$ and centered at $(0,1/n)$ for all $n=1,2,3,\ldots$.
I need to find a homeomorphism between $X$ and $H$.
Note: I am studying general topology, so I can't use algebraic topology.
Best Answer
You just map $\left[\frac12,1\right]$ onto the largest circle, $\left[\frac13,\frac12\right]$ onto the largest one and so on…
That is, if $x\in(0,1]$, then $x=\frac\lambda{n+1}+\frac{1-\lambda}n$ for some $n\in\mathbb N$ and some $\lambda\in[0,1]$. Then$$f(x)=\left(\frac1n\sin(2\pi\lambda),\frac1n-\frac1n\cos(2\pi\lambda)\right).$$And, of course, $f(0)=(0,0)$.