Convergent sequences in the cofinite topology

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SUMMARY

The discussion centers on identifying convergent sequences within the cofinite topology. It is established that any sequence that does not oscillate between two values can converge to a point in this topology. Constant sequences converge to their respective constant values, while divergent sequences approaching positive or negative infinity converge to all points in the space. The distinction between sequences with finite and infinite values is crucial for understanding convergence in this context.

PREREQUISITES
  • Cofinite topology concepts
  • Understanding of convergence in topological spaces
  • Knowledge of sequences and their properties
  • Familiarity with finite and infinite sets
NEXT STEPS
  • Research the properties of the cofinite topology
  • Study examples of convergent and divergent sequences
  • Explore the implications of sequences with finite versus infinite values
  • Learn about other topological spaces and their convergence criteria
USEFUL FOR

Mathematicians, students of topology, and anyone interested in the properties of sequences within different topological frameworks.

GridironCPJ
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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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Try to differentiate between:

The sequence takes on a finite number of values

and

The sequence takes on an infinite number of values.
 

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