As is described in the title, I believe $ \{ y = x/n : n \in \mathbb N+ \}$ is homeomorphic to the infinite wedge sum $\bigvee _\infty \mathbb R $, since the natural bijection is continuous at the crossing point in both direction. But a friend of mine told me it was wrong.
Another related question which appears on Hatcher's text is the union of circles centered $(n,0)$ with radius $n$. Again, it is claimed that it is not homeomorphic to the infinite wedge $\bigvee _\infty S^1 $, and I can't figure out the reason.
Could anybody explain the two baffling questions please? Thanks!!
In both cases the subspace topology that your union inherits from the plane is metrizable. The corresponding wedge, a quotient of the union, is not. At the vertex the quotient does not have a countable neighbourhood base: given a countable sequence $\langle U_n:n\in\mathbb{N}\rangle$ of neighbourhoods take in the $k$th space a neighbourhood $O_k$ of the point corresponding to the vertex that is a proper subset of $\bigcap_{n\le k}U_n$. Then the $O_k$ determine a neighbourhood $O$ of the vertex that contains none of the $U_n$.