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

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 short, the neighborhoods of the origin for both subspaces of $\mathbb{R}^2$ are "uniform"-they contain a uniformly positive length from every component of the union. This doesn't hold for the wedge sums. Nov 19, 2018 at 17:42

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 $$U_n$$.