Topologies on the Real Line
My book lists ten collections of subsets of R (the set of all real numbers), and tells me that exactly three of them are topologies on R. My problem is that nine of the ten look like topologies to me. Can somebody tell me what Iím missing?
This one is not a topology, as it seems to me, because for example
(itís a disjoint set). But what about the other nine (I will list them below)? I suspect that the three Iím supposed to pick are ii, iii and v, which involve only open intervals, but I donít see why a collection of closed or half-open sets couldnít still meet the definition of a topology.
Here are the candidates:
Re: Topologies on the Real Line
Consider the geometric sum . Let . Then the interval for all . Consider . Since for any , the union must be the set . This is why closed and half-closed intervals don't work for topologies. Unions are not closed.
For , let be the sequence . Hence, is a strictly increasing sequence of rational numbers. The union . So, that doesn't work, either.
The three that are topologies are (I think)