Supplementary Exercises*: Nets: Problem 1 Solution »

Supplementary Exercises*: Nets

  1. A directed set is a partially (strictly) ordered set such that for any pair of elements there is such that and .
  2. We can require it to be a strict partial order or a weak partial order: the definitions and results below hold in either case
  3. A cofinal subset of a partially ordered set is a subset such that for any element in there is a greater element in .
  4. If is a directed  set then its cofinal subsets are directed as well.
  5. A net in a topological space is a function from a directed set  into  : where .
  6. In other words, a net is a subset of the space indexed by a directed set.
  7. A sequence is a net.
  8. A subnet of the net is where is such that and is cofinal in .
  9. A subsequence is a subnet, but not every subnet of a sequence is a subsequence.
  10. In fact, if a subnet was defined as a cofinal subset of a net then every subnet would be a subsequence, but then this definition would not be "flexible" enough for some criteria stated below. For example, a sequence may have a convergent subnet but no convergent subsequence.
  11. The net converges to a point if for each neighborhood of there is such that implies . We denote this by .
  12. If is Hausdorff then a net cannot converge to more than one point.
  13. If , then .
  14. A net converges to iff every its subnet converges to .
  15. An accumulation point of net is a point such that for every its neighborhood the subset of indexes such that is cofinal in .
  16. A net has an accumulation point iff it has a subnet converging to .
  17. The closure and nets.
  18. iff there is a net of points of converging to .
  19. Compare to the Sequence Lemma of Section 21:
  20. if there is a sequence of points of converging to then the sequence is a net in and
  21. but if then there is just a net in converging to and this does not guarantee the existence of a sequence of points of converging to : something we can show if the space is metrizable
  22. Continuous functions and nets.
  23. is continuous iff for every convergent net : .
  24. Compare to Section 21:
  25. if is continuous then the image of any net converging to any is a net (indexed by the same directed set) converging to , in particular, this is true for any convergent sequence of points of
  26. but if the image of any sequence converging to any converges to , this can still be not enough for the continuity of : it is enough if the space is metrizable
  27. Compactness and nets.
  28. is compact iff every net in has a convergent subnet.
  29. Compare to Section 28:
  30. if a space is compact then every sequence has a convergent subnet, but this does NOT imply that the sequence has a convergent subsequence
  31. also if every sequence has a convergent subsequence this implies that every sequence has a convergent subnet but does not imply that every net has a convergent subnet
  32. given these two, compactness and sequential compactness are not comparable
  33. Let and be the nth digit of the binary expansion of (no tails of 1’s). is compact but not sequentially compact.
  34. For any subsequence let be such that its ’s digit of the binary expansion is . Then does not converge.
  35. Let , . Then is not empty. Every point in is an accumulation point of the sequence. And vice versa. One cannot give an explicit example of such a point.
  36. Topological groups (an application):
  37. If is a closed subset and is a compact subset of a topological group then is closed in .