Chapter 2 Notes
Here are some of the things I gave in Chapter 2 that are different from
the text's development. Most notably, I introduced sequences (which
don't occur until Chapter 3 in the text), and gave substantially
different proofs of the Bolzano-Weierstrass Theorem and the Heine-Borel
Theorem.
- I defined the closure of a set E to be the intersection of the
closed sets containing E, and then proved that this is equivalent to the
text's definition.
- A point x is in the closure of a set E if and only if every nbd of
x intersects E, if and only if there is a sequence in E converging to x.
- A point x is a subsequential limit of a sequence {x_n} if and only
if every nbd of x contains x_n for infinitely many n, if and only if x
is in the intersection over k in N of the closures of the sets
{ x_n : n >= k }.
- I didn't mention (but you should know) the definition of
*relatively open* set, and the fact that compactness is independent of
the ambient metric space (unlike openness or closedness, for example).
- I gave some conditions equivalent to a set A in a metric space
being bounded:
- there exists x in X and M in R such that d(x,y) <= M for all y in A;
- there exists x in X such that sup { d(x,y) : y in A } < infinity;
- for all x in X, sup { d(x,y) : y in A } < infinity;
- sup { d(x,y) : x,y in A } < infinity (and this latter sup is
called the "diameter" of A).
- Every sequence of real numbers has a monotone sequence.
- Our version of the Bolzano-Weierstrass Theorem: "Every bounded
sequence in R^n has a convergent subsequence". We'll show later that
this is equivalent to the text's version (involving limit points of
infinite sets).
- A set E is called *sequentially compact* if every sequence in E has
a subsequence converging to an element of E.
- In any metric space, compact => sequentially compact => closed and
bounded.
- R^n is *separable*, i.e., there is a countable dense subset (e.g.,
the points with rational coordinates).
- R^n is *second-countable*, i.e., there is a countable base for the
topology. A *base* for the topology of a metric space X is a collection
B of open sets in X such that every open set in X is a union of sets
from B. One of the simplest examples of a countable base for R^n is the
family of open balls with rational radii and centers with rational
coordinates.
- Lindelof Theorem for R^n: "For every family { U_t : t in A } of open
sets in R^n (indexed by a set A), there is a countable subset B of A
such that the union of U_t over t in B equals the union of U_t over all
t in A." We used this in the proof of the Heine-Borel Theorem.
- Our version of the Heine-Borel Theorem: "In R^n, closed and bounded
=> compact." The reason this is equivalent to the text's version is that
for any subset A of a metric space X, the following two statements are
true:
- Every sequence in A has a convergent subsequence in X <=>
every infinite subset of A has a limit point in X;
- A is sequentiallly compact <=> every infinite subset of A has a
limit point in A.