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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!!

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  • $\begingroup$ 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. $\endgroup$ Nov 19, 2018 at 17:42

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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 $U_n$.

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