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I have a conceptual question.

In a project I am working on, we are dealing with [tex]\Re[/tex][tex]^{n}[/tex] (with the usual topology), and I am working on characterizing some objects. In particular, I am dealing with the intersection of two simply connected open sets (that do not have any sort of pathological nature). I am identifying this intersection as the union of connected components.

My advisor asked me about the notion of connected components and why we care, since the intersection of open sets should always consist of connected components...I didn't have a good answer, and upon looking in Munkres, I don't have any better idea.

So, why do we consider connected components? It seems that every set is the union of connected components? I assume this notion would be more useful in the general setting rather than[tex]\Re[/tex][tex]^{n}[/tex] with < | >[tex]_{2}[/tex]?

I am studying engineering so please forgive my mathematical immaturity.

Thanks,

Mike