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If I want to consider the family $\{U_{a}\}_{a\in J}$ of path-connected subsets of a topological space $X$, can I assume they are pairwise disjoint? This would seem intuitively true. I need to prove that the homotopy classes $[\{-1,1\},\{1\};X,\{x\}]$ are in bijective correspondance with the $U_{a}$'s. I think I have an idea if I can assume that the $U_{a}$'s are disjoint!

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Do you want path components‌​? Otherwise, you can say something silly like, "I have $[0, 1]$, and $[0, 1)$ and are $(0, 1]$ are distinct path-connected subsets that overlap." – Dylan Moreland Feb 2 '12 at 20:18
In the statement of the problem it says components. I didn't read into the terminology too much. I should have looked up the term. :S Thanks again! – roo Feb 2 '12 at 20:22

Yes, two path components are disjoint. The key point is that the union of two path-connected sets with a common point (say, $p$) is also path-connected. Indeed, one can go from any point of one set to $p$ and from there to any point of another set.

So, if two path components had a common point, their union would be a strictly larger path-connected set, which is a contradiction.

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