- APPROXIMATE ISOMETRIES, COMPACTNESS OF MAPS, DIRECT LIMITS 157
where x.rv y if x E Ak and y E Ae for some k,£ EN and either fke (x) = y
(if k :::; £) or fek (y) = x (if£ :::; k). The relation rv is an equivalence relation.
The topology on limAk ____, is the quotient topology.Note that direct limits can be defined for more general directed systems,
but this is sufficient for our needs.
Let lg : Ae '---+ IhAk denote the inclusion map and 7f : IIkAk ---+ limAkdenote the quotient map and define
(4.5) le ~ 7f o l£ : Ae ---+ limAk, ____,the inclusion map into the direct limit. The topology on limAk ____, is the finest
topology such that the maps le : Ae ---+ limAk ____, are continuous for all£ E N.
Since the maps fk are one-to-one for all k EN, the maps fke are one-to-one
for all k :::; £. This implies the following.
LEMMA 4.13. The maps le· are one-to-one for all£ E N, and for all
m :2:: £we have
le= Imo fem·
PROOF. The first statement is obvious. The second statement follows
from lg (xe) rv lm Uem (xe)) (in Definition 4.12, we have suppressed the iden-
tification of x E Ae with its image lg (x) E IIkAk)· D
The following facts about direct limits are elementary.
LEMMA 4.14 (An open cover for the direct limit). If Ue C Ae is an open
set, then le (Ue) C limAk ____, is open, i.e., le are open maps. Thus {le (Ae)}£ENforms an open cover of limAk. ____,
PROOF. Since the fk are open maps for all k, we have that
7f-l [Ie (Ue)] = LJ fem (Ue) U LJ Ume)-^1 (Ue)
m?.£ m<£is open in IIkAk. Hence le (Ue) c limAk ____,. is open. D
This has implications about the structure ~f compact sets in the limit.COROLLARY 4.15 (Compact sets in the direct limit). If K C limAk ____,
is compact, then for k large enough, K. = Ik (Kk) for some compact set
K!C c Ak.
PROOF. K is covered by {le (Ae)}£EN and since it is compact, it is also
covered by a finite number of these. Since le (Ae) C Im (Am) if£ :'.Sm, we see
that K is in the image of Ik for large enough k: Since Ik is a homeomorphism
onto its image, Kk = IJ;^1 (K) is compact. D
Recall that a topological space is called second-countable if its topol-
ogy has a countable base.