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!
Decomposing a space into path connected components
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general-topology
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4Do you want [path _components_](http://en.wikipedia.org/wiki/Connected_space#Connected_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." – 2012-02-02
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0In 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! – 2012-02-02