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Can we expect that we might apply notions similar to "simple connectedness" and "multiple connectedness" to the $p$-adic setting, in spite of the fact that the standard topology of $\mathbb{Q}_p^n$ is totally disconnected. Hope this question make some sense.

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This question strikes me as a little vague. What would you want such a notion to accomplish? –  Qiaochu Yuan Oct 25 '11 at 17:14
    
My idea is that although the topology is totally disconnected, one might still be able to define the notion of a loop (a closed curve!), and hence of homotopy and other notions as well. What I want ultimately is to show that there cannot be an analytic isomorphism from a ball in $\mathbb{Q}_p^n$ and punctured ball i.e. a ball with one smaller ball removed. –  user17090 Oct 25 '11 at 17:21
    
@Ali: Have you considered applying the technology of algebraic geometry? –  Zhen Lin Oct 25 '11 at 18:22
    
@Zhen: I don't see exactly how the technology of algebraic geometry would help. Do you have an idea in the case at hand, i.e. trying to prove there cannot be an analytic isomorphism from a ball onto a punctured ball? –  user17090 Oct 25 '11 at 19:21
    
@Ali: There are similar problems in algebraic geometry: the Zariski topology does not accurately reflect the geometric properties of the objects we wish to study, so we have to construct replacements for the usual topological constructions. For example, there is a notion of an étale fundamental group, which substitutes for the usual fundamental group. –  Zhen Lin Oct 25 '11 at 19:27

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