CompuChip said: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 [itex]\cup[/itex] \{ x \}) and B. Then A' and B are open sets in Rn and their union is Rn, proving that Rn is disconnected.
In mathematics, a set is considered connected if it cannot be divided into two non-empty subsets that are both open and disjoint. In other words, there are no breaks or gaps in the set and all points are reachable from any other point.
Proving that R^n\{x} is connected is important for various reasons. One of the main reasons is that it helps to understand the structure and properties of the set. It also allows for the application of various mathematical theorems and concepts that rely on the connectedness of a set.
The key steps in proving the connectedness of R^n\{x} involve showing that the set is both open and closed, and that it cannot be divided into two disjoint subsets. This is often done by assuming the contrary and using proof by contradiction to arrive at a contradiction.
Yes, there are certain special cases and exceptions when proving the connectedness of R^n\{x}. One common exception is when the set is a single point, in which case it is trivially connected. Another exception is when the set is a line or curve, which may require a different approach to prove its connectedness.
The concept of connectedness is applicable in various fields such as physics, engineering, and computer science. For example, in physics, connectedness is used to study the behavior of particles and their interactions in a system. In engineering, it is used to analyze the stability and strength of structures. In computer science, connectedness is applied in network analysis and data clustering algorithms.