# Thread: show the space is connected

1. ## show the space is connected

For $n = 1,2,..,$ let $X_n = \{ \left( {1/n,y} \right) - n \le y \le n\}$Show that the subspace $R^n - \cup _n X_n$ is connected.

2. ## Re: show the space is connected

Originally Posted by rqeeb
For $n = 1,2,..,$ let $X_n = \{ \left( {1/n,y} \right) - n \le y \le n\}$Show that the subspace $R^n - \cup _n X_n$ is connected.
This has to be a proof by contradiction. Suppose that $R^n - \cup _n X_n$ 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.