Is the following conjecture true ? I am trying to collect some properties about box topology and product topology in order to study for topology exam.
is the product of connected compact space connected in the box topology?
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No: see this question and its answer. – Brian M. Scott Nov 17 '15 at 08:47
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Could you give me explicit example ? – Illustionist Nov 17 '15 at 08:51
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I guess one example would be $[0,2]^{\omega}$ that is not connected right ? – Illustionist Nov 17 '15 at 09:24
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@BrianM.Scott did you notice compact ? – Illustionist Nov 17 '15 at 09:35
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Yes. Just replace $\Bbb R$ by $[0,1]$; essentially the same argument works. – Brian M. Scott Nov 17 '15 at 10:12
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But using his argument why is then $[0,1]^{\omega}$ not connected ? – Illustionist Nov 17 '15 at 10:39
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If the space were connected, it would be the component of every point. The argument shows that this is not the case: the component of a point is very much smaller than the whole space. Therefore the space is not connected. – Brian M. Scott Nov 17 '15 at 10:41
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ok that is great I understand now ! – Illustionist Nov 17 '15 at 10:50