# On topology of p-adic numbers. [closed]

This may be a stupid question.But I am stuck with it.Is Q_p(the p-adic) connected under the usual topology?I was confounded with this problem while trying to construct a counter-example related to my master's thesis.

## closed as off topic by George Lowther, Chandan Singh Dalawat, user9198, Tony Huynh, Marc PalmDec 19 '12 at 10:03

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• Both Wikipedia and Google know the answer to this (very elementary) question. If you don't find it there or by yourself, you should try math.stackexchange.com – Olivier Dec 19 '12 at 9:34
• No, $\mathbf{Q}_p$ is about as far from being connected as possible: it's totally disconnected - the maximal connected subsets are points. – David Loeffler Dec 19 '12 at 9:35
• $\mathbb{Q}_p$ is a punctured Cantor set, meaning that it is homeomorphic to a Cantor set minus one point. – Lee Mosher Dec 19 '12 at 14:39

Any non-empty open can be written as a disjoint union of opens ; for example $\mathbb{Z}_p=\cup(a+p\mathbb{Z}_p)$ where $a$ runs through $\{0...(p-1)\}$. Those spaces are said totally disconnected.