Is *R Connected in the S-Open Topology?

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SUMMARY

The hyperreals, denoted as *R, are confirmed to be totally disconnected with respect to the standard topology of open intervals. In the context of the S-open topology, which utilizes open balls defined as ((x-r, x+r)), *R remains disconnected. The discussion concludes that *R does not possess connected components under the S-open topology, reinforcing the notion of total disconnectedness in hyperreal analysis.

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jem05
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Hello,

I know the hyperreals are not connected with respect to the interval opens (open sets are open intervals) In fact *R is totally disconnected.
Is *R connected with respect to the S-open topology (open ball around x of radius r is ((x-r, x+r)) any point in this set has its halo in the set also. In other words, you take the interval (x-r, x+r) and take out the points of infinitessimal distance with the endpoints. You are then left with ((x-r, x+r)).
If *R is not connected, what are the connected components?

Thank you
 
Physics news on Phys.org
ok, got it.
it actually is not connected
 

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