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Consider a continous map $f:Y \to X$ between topological spaces. The pre-cosheaf $\mathcal{F}: Open(X) \to Set$ of connected components of the inverse image is defined as $\mathcal{F}(U):= \pi_0(f^{-1}(U)), U\subset X$ open.

In case that Y is not locally-connected: Why can $\mathcal{F}$ fail to be cosheaf?

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  • $\begingroup$ If Y is not locally connected, how is π_0(f^{−1}(U)) defined? Connected components of a non-locally-connected topological space do not in general form a set, but only a pro-set. If you replace sets with pro-sets in your question, the resulting functor π_0 is cocontinuous and F is a cosheaf, see ncatlab.org/nlab/show/pro-set#the_proset__of_a_locale. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 12, 2016 at 15:35
  • $\begingroup$ @Dmitri Pavlov "Connected components of a non-locally-connected topological space do not in general form a set, [...]" Why not, thanks? $\endgroup$
    – Jo Wehler
    Commented Mar 12, 2016 at 16:00
  • $\begingroup$ A connected component is an open subset with an open complement that does not have a nonempty proper subset with the same property. There are plenty of topological spaces such that for every open subset one can find a proper open subset with this property, for example, the Cantor set. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 12, 2016 at 17:52
  • $\begingroup$ @Dmitri Pavlov In my opinion your definition is not equivalent to the general definition, e.g., Bourbaki, General Topology, Chap. I, §11.5, Def. 3. The components of the set of rationals are singletons, notably they are not open. $\endgroup$
    – Jo Wehler
    Commented Mar 12, 2016 at 20:25
  • $\begingroup$ I see, so you are really looking at the limit of the pro-set discussed above, which computes the same set as Bourbaki's definition, at least for compact topological spaces. As discussed by Porter in Čech homotopy II, taking the limit tends to destroy (co)descent properties, unless the topological space satisfies a fairly strong stability condition. Presumably, the example of an unstable space discussed there (specifically, the dyadic solenoid) should then give an example for which the codescent property fails. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 12, 2016 at 23:20

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