Union of lines $ \{ y = x/n : n \in \mathbb N+ \}$ not homeomorphic to infinite wedge sum of lines?

Question

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!!

Answer 1

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 $U_n$.