Proving R^n\{x} is Connected for n>1

[R.P.F.]

Homework Statement

I am trying to show that R^n\{x} where x could be any point in R^n is connected for n>1. I have been thinking about this for a while but haven't had any luck. Seems like that I'm missing something simple. I tried to construct a continuous map from R^n to R^n\{x}. All I came up was to send x to some other point y, but that map did not seem to be continuous. Any help is appreciated!

Homework Equations

The Attempt at a Solution

 

[CompuChip]

Isn't it easier to show that it is path-connected, which implies connected? You can easily find a path between any two points (take a straight line, if it crosses x then make a wobble around it) - I'll leave it up to you to formalise that argument.

If you insist on using the definition, maybe a proof by contradiction would work. I'm thinking along the lines of "let A and B be open disjoint sets in X = Rⁿ \ {x} whose union is X. Let A be the one containing an open neighbourhood of x, and consider A' = (A $\cup$ \{ x \}) and B. Then A' and B are open sets in Rⁿ and their union is Rⁿ, proving that Rⁿ is disconnected.

 

[R.P.F.]

Isn't it easier to show that it is path-connected, which implies connected? You can easily find a path between any two points (take a straight line, if it crosses x then make a wobble around it) - I'll leave it up to you to formalise that argument.

If you insist on using the definition, maybe a proof by contradiction would work. I'm thinking along the lines of "let A and B be open disjoint sets in X = Rⁿ \ {x} whose union is X. Let A be the one containing an open neighbourhood of x, and consider A' = (A $\cup$ \{ x \}) and B. Then A' and B are open sets in Rⁿ and their union is Rⁿ, proving that Rⁿ is disconnected.

Hey,

Thanks for your ideas! Using the second idea you mentioned, I was able to write up a proof using the definition. The proof was not as simple I expected. So yeah, now I'm going to do the path-connected business. :rofl: