The discussion centers on the relationship between path-connected spaces and their closures in topology. It is established that while a path-connected space X may intuitively suggest that its closure is also path-connected, this is not necessarily true. A key counterexample provided is the set {(x, sin(1/x)) | x ∈ ]0,1]} in ℝ², which is path-connected, but its closure is not. This example is frequently encountered in topology exams and serves as a critical reminder of the nuances in connectedness properties.
PREREQUISITESMathematics students, particularly those studying topology, educators preparing for exams, and anyone interested in the intricacies of connectedness properties in mathematical spaces.
micromass said:Ah, this is a very good question. In fact, this very question was asked on my topology exam...
It is maybe obvious to you (and to me), but it is false. Consider
\{(x,\sin(1/x))~\vert~x\in ]0,1]\}\subseteq \mathbb{R}^2
This is clearly a path connected set. However, it's closure is not path connected.
I suggest that you remember this example very well. Because it is a very frequent counterexample to all sort of connectedness-properties. And it's a popular exam question as well