1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

158 4. PROOF OF THE COMPACTNESS THEOREM

COROLLARY 4.16 (Second-countable direct limits). If each Ak is the

countable union of compact sets, then so is limAk. In particular, limAk is

---+ ---+

second-countable.

()()

PROOF. If Ak = U Kke where Kke is compact, then

£=1

()() ()() ()()

limAk = LJ h (Ak) = LJ LJ Ik (Kke)

---+ k=l k=l e=l

which is a countable union of compact sets.

LEMMA 4.17 (Direct limit of Hausdorff spaces is Hausdorff).

(1) If x E limAk, there exist£ and xe E Ae such that le (xe) = x.

---+

(2) If each Ak is Hausdorff, then limAk is Hausdorff.

---+

D

PROOF. (1) Since x E limAk, there must be xe E IIkAk such that
---+
7r (x.e) = x, and thus xe E Ae for some£.
(2) Given x i= y E limAk, there exists Xk E Ak and Y.e E Ae for some
---+

k, £ E N such that h (xk) = x and le (y.e) = y. Assume without loss of

generality that £ 2: k. Define x.e ~ fke (xk) E Ae. Since x i= y, we have
xe i= ye. Since Ae is Hausdorff, there exist disjoint open neighborhoods Nx
and Ny of xe and Ye, respectively. Since le is one-to-one and open, we
conclude that le (Nx) and le (Ny) are disjoint open neighborhoods of x and
y, respectively, in limAk. D
---+
The direct limit satisfies the following universal property (see Proposi-
tion 15.3 in [159]). For any space X and maps 1/Jk : Ak ~ X such that

1/Je o fke = 1/Jb

there exists a unique map '11 : limAk ~ X such that
---+
1/Jk = '11° h.

3. Construction of good coverings by balls

3.1. Overview. In this section we shall prove the following lemma,
which we will need in order to construct maps between different manifolds.
The reader is warned that in this section we will be using superscripts which
usually do not represent exponents. We shall need five radii of balls so that
the smallest are disjoint and so that all others cover and are successively
larger to allow for maps between the intersections. This section is quite.
technical and the proofs may be skipped in the first reading.

LEMMA 4.18 (Existence of good coverings by balls). There exist a sub-

sequence of {(Mk, 9k, Ok)} , convex geodesic balls Bk C Bk C Bk C Bk C