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Suppose $X$ is a locally path connected topological space and $\mathcal{U}$ is an open cover of $X$ (consisting of path connected sets if we want).

One often wants the intersection $A\cap B$ of pairs of elements $A,B\in \mathcal{U}$ to be path connected, or perhaps stronger, that the intersection of finitely many elements of $\mathcal{U}$ be path connected. This, for instance, is the case in some groupoid versions of the van Kampen theorem (like the one in Peter May's A Concise Course in Algebraic Topology). Having covers with this property also simplifies life a bit in the study of shape invariants constructed via nerves of covers. If we don't start with a cover this nice perhaps we can at least get to one by refinement.

Question: Is it always possible to find an open cover $\mathcal{V}$ of $X$ refining $\mathcal{U}$ such that the intersection of every pair of elements in $\mathcal{V}$ is path connected (or empty)? Can we do even better and find $\mathcal{V}$ such that the intersection of finitely many elements of $\mathcal{V}$ is path connected?

I'm less confident such refinement is possible for general locally path connected $X$. I'd be perfectly content to assume $X$ is paracompact Hausdorff.

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2 Answers 2

If $X$ is a smooth manifold, there is always a refinement to a given open cover such that any finite intersection is connected. Namely, choose a complete Riemannian metric on $X$ and choose a refinement by geodesically convex set.

One can surely push this property to somewhat more general spaces like metric spaces which locally admit unique paths realizing the distance. Maybe, simplicial complexes can be equipped with a metric with this property.

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This might be considered in the Meyer-Vietoris Sequence as some general form of Van's Thm.

And the question may be reduced to ask a stronger version, that whether $H_{0}(A\cap B)$ is trivial for any pair in the cover U. And in the corresponding M-V Seq, this reduces to finding a set of refinement that each of such pair A,B intesecting X must satisfy $H_{0}(A)\oplus H_{0}(B)$ is trivial, which requires a stronger condition on the cover V.

So I guess your wish is hardly possible since you assume it in a c-Hausdorff space, where path-connected must lead to arc connected.

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I am doubtful that restating my question in terms of 0-th homology is helpful. –  Jeremy Brazas Mar 31 '13 at 13:59
    
I present my thoughts about why I believe this restatement is helpful in edited answer. –  Henr.L Apr 1 '13 at 5:00
    
The cone over any space is simply connected and moreover is contractible. It has nothing to do with path connectivity. –  Jeremy Brazas Apr 1 '13 at 12:29
    
You're right. Lack of consideration. –  Henr.L Apr 2 '13 at 3:51

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