For $\displaystyle n = 1,2,..,$ let $\displaystyle X_n = \{ \left( {1/n,y} \right) - n \le y \le n\} $Show that the subspace$\displaystyle R^n - \cup _n X_n $ is connected.
For $\displaystyle n = 1,2,..,$ let $\displaystyle X_n = \{ \left( {1/n,y} \right) - n \le y \le n\} $Show that the subspace$\displaystyle R^n - \cup _n X_n $ is connected.
This has to be a proof by contradiction. Suppose that $\displaystyle R^n - \cup _n X_n $ is not connected. Then it is the union of two open disjoint sets U and V.
Just to get you started, think about the fact that the origin must lie in one of those sets.