Convergent sequences in the cofinite topology

In summary, a convergent sequence in the cofinite topology is a sequence of elements in a set where the elements get closer and closer to a specific value, which is the limit of the sequence. The cofinite topology is different from other topologies because it only considers sets that are finite or have a finite complement. This means that the open sets in the cofinite topology are all sets that contain all but a finite number of elements from the original set. In the cofinite topology, a sequence can only converge to one limit and it cannot converge to multiple limits. The cofinite topology is coarser than the discrete topology, but the discrete topology has more open sets, making it finer. A sequence in the cofinite topology can diverge
  • #1
GridironCPJ
44
0
How can you identify the class of all sequences that converge in the cofinite topology and to what they converge to? I get the idea that any sequence that doesn't oscillate between two numbers can converge to something in the cofinite topology. Considering a constant sequence converges to the constant, a divergent sequence to +- infinity converges to all points, a sequence that gets infinitely closer to a number converges. Am I essentially on the right track here or can anyone give me a counterexample to my claim?
 
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  • #2
Try to differentiate between:

The sequence takes on a finite number of values

and

The sequence takes on an infinite number of values.
 
  • #3
Yes, you are essentially on the right track. In the cofinite topology, a sequence converges if and only if it eventually stays within any open set, i.e. it eventually stays outside of only finitely many points. This means that any sequence that does not oscillate between two numbers, or more generally, any sequence that eventually stays within a finite set of points, will converge in the cofinite topology.

However, there are some counterexamples to your claim. For example, consider the sequence (1, 2, 3, ...) in the cofinite topology on the integers. This sequence does not oscillate between two numbers, but it does not converge to any point in the topology. This is because for any open set containing a point n, there are infinitely many points in the sequence that are not in the set, and therefore the sequence does not eventually stay within the open set.

Another counterexample is the sequence (1, 1/2, 1/3, ...) in the cofinite topology on the real numbers. This sequence gets infinitely closer to 0, but it does not converge to 0 in the topology. This is because for any open set containing 0, there are infinitely many points in the sequence that are not in the set, and therefore the sequence does not eventually stay within the open set.

In general, any sequence that eventually stays within a finite set of points will converge in the cofinite topology, but there may be sequences that do not oscillate between two numbers and do not eventually stay within a finite set of points, and thus do not converge in the topology.
 

1. What is a convergent sequence in the cofinite topology?

A convergent sequence in the cofinite topology is a sequence of elements in a set where the elements get closer and closer to a specific value. In the cofinite topology, this value is the limit of the sequence.

2. How is the cofinite topology different from other topologies?

The cofinite topology is different from other topologies because it only considers sets that are finite or have a finite complement. This means that the open sets in the cofinite topology are all sets that contain all but a finite number of elements from the original set.

3. Can a sequence converge to multiple limits in the cofinite topology?

No, a sequence can only converge to one limit in the cofinite topology. This is because the open sets in the cofinite topology contain all but a finite number of elements, so as the sequence gets closer to a specific value, it will eventually be contained in an open set that does not contain any other potential limit points.

4. What is the relationship between the cofinite topology and the discrete topology?

The cofinite topology is coarser than the discrete topology, meaning that the open sets in the cofinite topology are also open sets in the discrete topology. However, the discrete topology has more open sets, making it finer than the cofinite topology.

5. Can a sequence in the cofinite topology diverge?

Yes, a sequence in the cofinite topology can diverge if it does not have a limit. This can happen if the sequence does not get closer and closer to a specific value or if the sequence oscillates between multiple values.

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