Section 28: Limit Point Compactness

1. A limit point compact space (Bolzano-Weierstrass property, Fréchet compact, weakly countably compact) is a space such that every its infinite subset has a limit point.
2. A sequentially compact space is a space such that every sequence of points has a convergent subsequence.
3. A countably compact space is a space such that every countable open covering has a finite subcovering.
4. A space is countably compact iff every countable nested sequence of closed nonempty subsets has a nonempty intersection.
5. note that here the non-emptiness of the intersection of any nested collection of closed sets is not only necessary but sufficient as well
6. Relations:
7. any space:
8. compact or sequentially compact $\Rightarrow$ countably compact $\Rightarrow$ limit point compact
9. take an infinite sequence of nested closed sets; if the space is compact then the intersection is not empty; if the space is sequentially compact then take a point in each set, there is a convergent subsequence and its limit must lie within the intersection
10. any countable subset that contains no its own limit points can be covered by a countable number of open set containing one point of the subset only; there is no finite subcovering; if the space is countably compact, then the subset is not closed and has a limit point
11. in general, there is no relation between compactness and sequential compactness.
12. the first uncountable ordinal is not compact but it is sequentially compact
13. the uncountable product of the unit intervals is compact but it is not sequentially compact
14. a $T_1$ -space:
15. limit point compact $\Rightarrow$  countably compact
16. take an infinite sequence of nested closed sets and a point in each set; if the set of all points is finite, then the intersection is not empty, otherwise, there is a limit point; if the limit point is outside the intersection, then there is its neighborhood intersecting only finitely many sets of the collection; now we use the T₁-property to construct another neighborhood that does not contain any points of the sequence different from the limit point
17. a first-countable space (Section 30):
18. countably compact $\Rightarrow$ sequentially compact
19. take a sequence; suppose some point of the space is such that every its neighborhood contains a subsequence; now we use the first-countability: if there is a countable basis {B_n} at the point, then for every n take a point of the sequence contained in B1∩...∩Bn with an index greater than the previous one — this subsequence has to converge to the point; now take the closure of the sequence; it is closed and, hence, countably compact; if all points in the closure have a neighborhood that does not contain any subsequence then there is no finite subcovering; therefore, some point in the closure is such that every its neighborhood contains a subsequence
20. a $T_1$ and first-countable space:
21. limit point compact $\Rightarrow$ sequentially compact
22. we may use the two previous statements but it is more useful to proof this directly: if a sequence contains a constant subsequence then the subsequence is convergent, otherwise there are infinitely many different elements in the sequence and it has a limit point; the T1 property guarantees that every neighborhood of the limit point contains a subsequence, and the first-countability ensures that we can construct a subsequence convergent to the limit point
23. a second countable space (Section 30):
24. countably compact $\Rightarrow$ compact and sequentially compact (all three are equivalent)
25. if there is a covering and a countable basis then take every basis open set that is contained in an open set of the covering and for each such basis set select one open set of the covering that contains it, this way we obtain a countable and then a finite subcovering
26. if the space is second-countable then it is first-countable
27. a metric space or a second-countable $T_1$ -space:
28. all four are equivalent
29. a metric space is first-countable and T1, therefore, (c implies cc, lpc and sc) and (cc iff lpc iff sc)
30. now, if we knew that the space is also second countable, then we would conclude that all four are equivalent, however, there are metric spaces that are not second-countable (for example, discrete uncountable space or a fancier one: take the unit square and suppose that one can move horizontally at the bottom of the square only), but...
31. if a metric space is limit point compact then it is second-countable
32. first, for every positive r there is a covering by a finite number of r-balls (if there is no finite covering by r-balls then there is an infinite set of points such that the distance between any two points is at least r but it has no limit points); second, take the union of all finite collections of 1/n-balls covering the space — it is a countable basis
33. all four are equivalent to the requirement that the space is bounded under every metric that induces the topology
34. see Exercise 3 of Section 35: if compact then the distance being continuous is bounded; if every metric is bounded and there is a continuous function then we can construct a metric such that it is bounded iff the function is bounded
35. idea: this, actually, provides another way to show that countable compactness of a metric space implies compactness, namely, if countably compact then must be bounded in any metric, therefore, compact
36. Examples:
37. $S_\Omega$ is not compact (no largest element) but it is limit point compact (an infinite set has a countable subset which is bounded and lies within a compact closed interval).
38. $\overline{S}_\Omega$ is compact and, therefore, limit point compact.
39. Subsets and limit point compactness:
40. A closed subset of a limit point compact space is limit point compact.
41. A limit point compact subset of a Hausdorff space does NOT have to be closed (even if the space is compact).
42. For example, $S_\Omega\subset\overline{S}_\Omega$ .
43. Continuous functions and limit point compactness:
44. The image of a limit point compact set under a continuous function does NOT need to be limit point compact.
45. Products and limit point compactness:
46. Even the product of two Hausdorff limit point compact spaces does NOT have to be limit point compact.
47. An isometry ($d(f(x),f(y))=d(x,y)$ ) from a compact metric space into itself is a homeomorphism (i.e. it must be surjective).
48. A shrinking map ($d(f(x),f(y))<d(x,y)$ ) from a compact metric space into itself has a unique fixed point: $f(x)=x$ .
49. A contraction ($d(f(x),f(y))\le\alpha d(x,y)$ for some $\alpha<1$ ) from a compact metric space into itself has a unique fixed point (this result for contractions only, i.e. not for shrinking maps in general, can be generalized to non-compact complete metric spaces, see Section 43).