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Appendix B Compactly generated spaces
Appendix B Compactly generated spaces In this Appendix B we study the category CG of compactly generated spaces which, as we have seen in Section 6.1, contains the category of CW-complexes; moreover, CG is used in the development of the section on F-fibrations (see Section 7.2). To define CG, we follow closely the approach taken by Norman Steenrod in [29]. We begin by recalling that in Section 1.1 we introduced the evaluation function E:
x x M ( X ,Y )+ Y
defined by ~ ( zf, ) = f ( z ) and we used it to compare the function indeed, we saw that if X spaces M ( X x 2,Y)with M(Z,M(X,Y)); is locally compact Hausdorff, then E is a map and M ( X x 2,Y)2 M ( Z , M ( X , Y ) ) .In this section we show how to get rid of the cumbersome hypothesis on X by restricting somehow the category of topological spaces on which we work; however, we shall proceed in such a way that the category we obtain - the category of compactly generated spaces - is still large enough to encompass many of the most important spaces we should like to deal with whenever working in Homotopy Theory. A space X is said to be compactly generated if X is Hausdorff and its topology is determined by the set of all its compact subsets, i.e., U C X is closed iff, for every C c S compact, U f l C is closed in C. The category of compactly generated spaces will be denoted by CG; notice that CG is a full subcategory of Top.
278
APPENDIX B. COMPACTLY GENERATED SPACES
CW-complexes are compactly generated spaces: if X is a CWcomplex, its closed cells are compact; moreover, the topology of X is determined by its closed cells (see Theorem 6.1.5). The following lemma gives a useful test to see if a space is compactly generated.
Lemma B . l Let X be a Hausdorff space. Suppose that f o r every subspace U c X and every x E U there exists a compact set C c X such Then X E CG'. that x E
m.
Proof - Let U be a subset of X such that, for every C c X compact, U n C is closed. Let be the closure of U ;for every x E D,there exists a compact subset C of X such that x E U n C = U n C . Hence, U =
u
0
r.
Corollary B.2 The category CG includes all the locally compact Hausdorff spaces and the spaces satisfying the first axiom of countabilaty (in particular, all metric spaces). Proof - Let X be a locally compact Hausdorff space. Given that U is an arbitrary subset of X and that 2 E 0, take a neighborhood V, of x whose closure is compact; then
v,
Now suppose that X satisfies the first axiom of countability. We take as a compact set C associated to x E U c X the set {xc) together with a sequence converging to 2. 0
Lemma B.3 Let f : X --+ Y be a function with X E Cg and Y E Top. Then f is continuous iff its restriction t o any compact subset of X is continuous. Proof - Let K be an arbitrary closed subset of Y ; then, for every compact subset C c X, f - ' ( K ) n C = f c ' ( K ) - where f c is the restriction of f to C - is closed by hypothesis. This shows that f is continuous. 0 Compactly generated spaces are indeed very much related to locally compact spaces, as we can see from the following result:
279 Theorem B.4 A Hausdorff space X is compactly generated iff it is the quotient of a locally compact Hausdorg space.
Proof -
j
Index all compact subsets of X in a set A and define the
surjection c:
u CA+X
XEA
to be the function whose restriction to any CX is the identity map. The topological sum C = UXEA CA is compactly generated and so, by Lemma B.3, c is continuous. Furthermore, C is locally compact. We need only prove that c is an identification map. In fact, if K c X is such that c - l ( K ) n CX is closed, for every A, then K is closed in X because c-l ( K )n CX= K n CA. + Suppose now that Y is locally compact Hausdorff and q : Y + X is an identification map. Let K c X be such that, for every X E A, K n Cx is closed in CA;we wish to prove that K is closed in X . Since Y is locally compact Hausdorff, every point y E Y has a neighborhood Vwwhose closure is compact; cover q - l ( K ) by these open sets:
q(c)
is closed in q(F)that is to say, Because q ( q ) is compact, K n there is a closed subset Kg of X such that
and consequently,
p ( K )n
= q-I(K,) n V ,
is closed in Y . This means that, for every compact subset C c Y , qW1(Kn ) C is closed in C. Since Y E CG, it follows that q - l ( K ) is closed in Y and therefore, K is closed. in X. 0
Corollary B.5 If 'I' E CG and q : Y + X is an identification map, then X E Cg.
280
APPENDIX B. C O M P A C T L Y GENERATED SPACES
Proof - By the theorem, there exists a locally compact Hausdorff space 2 and an identification map p : 2 -+ Y ; then the composition
is again an identification and once more by Theorem B.4, X is compactly generated. 0 The Cartesian product of two compactly generated spaces (with the product topology) is not necessarily a compactly generated space (there exists a celebrated example by C.H.Dowker in this respect - see [15,Example 2, Section 2.21). We circumvent this problem with the following stratagem. Firstly, denote by X x Y the usual Cartesian product; then, define the product X 8 Y in CG to be the space with the same underlying set as X x Y but with the compactly generated topology. Indeed, several times in the sequel we shall want to change the topology of a Hausdorff space X into the finer compactly generated topology; to indicate this change of topology in a more consistent manner, we shall denote the new topological space obtained from X with k ( X ) ; thus, with this convention, X 8 Y = k ( X x Y ) . At this point one should notice that for any Hausdorff space X , the new space k ( X )is also Hausdorff; moreover, the identity function 1s : k ( X ) 3 X is continuous.
Theorem B.6 If X is locally compact Hausdorff and Y E Cg, then
X @ Y Z X X Y . Proof - Let A c X x Y be such that, for every compact subset C C X x Y ,A n C is closed. We wish to prove that A is closed, i.e., that ( X x Y ) \ A is open. To this end, take arbitrarily a point (z,,yo) E ( X x Y ) \ A ; since X is locally compact Hausdorff, there exists an open set U of X containing zo and such that the closure g is compact. Since x {yo} is compact, A n ( g x {yo}) is closed. Hence there is an open set If c U such that 2, E V , is compact and
v
(V x
{yo})
nA
=8
(v
.
Let B be the projection into Y of A n x Y ) . If C c Y is compact, An(V x C) is closed in r/ x C and hence, is compact; but then, B n C is closed. Because Y E Cg, it follows that B is closed in Y . To complete
281 the proof, observe that (zo,y o ) belongs to the open set V x (Y \ B ) and that
V x (Y \ B )
c (X x Y ) \ A .
0
Observe that Dowker's example has consequences also for function spaces: in fact, if X consists of just two points and Y E Top, then M ( X , Y ) = Y x Y . Then, for every X , Y E CG, we take the function space of all maps from X into Y in CG to be
Y" = k ( M ( X ,Y ) ) (note that M ( X , Y ) is Hausdorff because Y is Hausdorff - see Exercise 1.1.2). Similarly, we define the function space of based maps from ( X , z o )E CG to ( Y , y o ) E CG to be
(Y,yo)(s'"o) = k ( M , ( X , Y ) ). We shall convene that from now on, whenever we talk about products or function spaces of compactly generated spaces, it is to those specific product or function spaces we defined above that we are referring to. The next theorem paraphrases Theorem 1.1.2; it is the so-called exponential law for CG.
Theorem B.7 For every X , Y,2 E CG, a function
as
continuous i f the function
Proof - As we have seen in Theorem 1.1.2, there are no restrictions on any of the spaces involved for the proof of the necessary condition. Thus, let us assume that is continuous. We first prove that, for every compact subset K of X, the restriction of f to K 8 2 is continuous. In fact, take the inclusion map i : K ~f X and its induced map i" : Y s --+ Y K and consider the commutative diagram of Figure B.1. The
f'
282
APPENDIX B. COMPACTLY GENERATED SPACES
Y FIGURE B . l evaluation function E is continuous because K is compact (see Theorem 1.1.1) and thus, f I K @ J Zis continuous. This shows that the restriction of f to any compact subset C c X @J 2 is continuous: in fact, the projection p q : X 8 2 3 X is continuous and C c p q ( C ) 8 2. Finally, f is continuous by Lemma B.3. 0
Corollary B.8 For every X , Y E Cg, the evaluation function
) f(z) is continuous. given by ~ ( z , f = Proof - The function
E:
is adjoint to the identity map
and so, is continuous by the sufficiency part of the previous theorem. 0
Lemma B.9 Let X , Y be given Hausdorfl spaces and let f : X + Y be a continuous function; let k( f) : k ( X ) k ( Y ) be the function which coincides with f at the set-theoretical level. Then k ( f ) is continuous. --f
Proof - It is enough to show that the restriction of k ( f ) to any compact subset C c k ( X ) is continuous. We begin by noticing that X and k ( X ) have the same compact subsets: If C c k ( X ) is compact, l.y(C) = C is compact in S ;now suppose that C c X is compact
283 and let C' be C with the relative topology from k ( X ) . The identity function 1~ : C' t C is continuous; we wish to prove that its inverse is also continuous and so, C' E C is compact in k ( X ) . Let K c C' be closed;, since K meets every compact subset of X in a closed set, K n C = K is closed in C proving thereby that C' + C is continuous. Now let C' c k ( X ) be compact. B y what we have just seen, the same set C with its topology in X is compact; since f is continuous, f (C) is compact in Y and so is f ( C ) ' in k ( Y ) .But then, the restriction k( f ) I C' is continuous as a composition of continuous functions, namely the identity maps lc, l f ( qand the map f I C. 0
Theorem B.10 For every X , Y , Z E CG, YxBz E (Y-')'. Proof - Let
B : M ( X @ Z , Y )+ M ( Z , M ( X , Y ) )
f.
be the function taking any map f : X @ Z -+ Y into its adjoint By Theorem B.7, 9 is a bijection. The sets W K ,of~ all maps f : X + Y such that f ( K ) C U ,for all K c X compact and U c Y open, form a sub-basis S for the open sets of M ( X , Y ) ;by the unbased version of Lemma 1.1.5, the sets &(L, W K , U ) of maps 9 :z
-
M ( X , Y ) 7 s(L)c w c u
c Z compact, form a sub-basis for the open sets of M(2, M ( X , Y ) ) . On the other hand, the sets
with L
W K ~= L (9 , ~E M ( X
2, Y ) I g ( K
L) c
u)
with K c X compact, L c 2 compact and U c Y open form a sub, K,~) basis for the open sets of M ( X @ 2,Y).Now, because g E F % ( L W iff 5 E WK@L,U, that
P ( F % ( L IT&[-) , = w/cc;;r.,c. ; Lemma B.9 then proves that
-
k ( 4 ) : y.v'&z is a homeomorphism. 0
1z
7.1-
(1
284
APPENDIX B. COMPACTLY GENERATED SPACES
EXERCISES B.l Let X and Y be given Hausdorff spaces. Prove that
k ( X ) @ k ( Y )2
qx x Y ) .
B.2 A subset A of X
E Top is said to be compactly closed if, for every compact Hausdorff space K and every map f : K -+ X , f - ' ( A ) is closed in K . A space X E Top is said to be compactly closed if every compactly closed subset of X is closed. Let CTop be the category of all compactly closed spaces. Prove that CG is a subcategory of C Top.
B.3 Let X and Y be compactly closed spaces. Prove that the space X @ Y , obtained by first taking the usual Cartesian product space X x Y and then refining its topology to the compactly closed topology, is a product in the category CTop.
B.4 A compactly closed space X is weak Hausdorflif the diagonal space A X is closed in X @ X . Prove that, X is weak Hausdorff iff for every compact Hausdorff space K and every map f : K + X , f ( K ) is closed in X.
B.5 Prove that the compactly closed weak Hausdorff spaces form a subcategory of CTop. Show that the former category has the following advantage over the latter: if ( X ,A ) is a pair of compactly closed weak Hausdorff spaces, so is X / A .
x x M ( X ,Y )+ Y
defined by ~ ( zf, ) = f ( z ) and we used it to compare the function indeed, we saw that if X spaces M ( X x 2,Y)with M(Z,M(X,Y)); is locally compact Hausdorff, then E is a map and M ( X x 2,Y)2 M ( Z , M ( X , Y ) ) .In this section we show how to get rid of the cumbersome hypothesis on X by restricting somehow the category of topological spaces on which we work; however, we shall proceed in such a way that the category we obtain - the category of compactly generated spaces - is still large enough to encompass many of the most important spaces we should like to deal with whenever working in Homotopy Theory. A space X is said to be compactly generated if X is Hausdorff and its topology is determined by the set of all its compact subsets, i.e., U C X is closed iff, for every C c S compact, U f l C is closed in C. The category of compactly generated spaces will be denoted by CG; notice that CG is a full subcategory of Top.
278
APPENDIX B. COMPACTLY GENERATED SPACES
CW-complexes are compactly generated spaces: if X is a CWcomplex, its closed cells are compact; moreover, the topology of X is determined by its closed cells (see Theorem 6.1.5). The following lemma gives a useful test to see if a space is compactly generated.
Lemma B . l Let X be a Hausdorff space. Suppose that f o r every subspace U c X and every x E U there exists a compact set C c X such Then X E CG'. that x E
m.
Proof - Let U be a subset of X such that, for every C c X compact, U n C is closed. Let be the closure of U ;for every x E D,there exists a compact subset C of X such that x E U n C = U n C . Hence, U =
u
0
r.
Corollary B.2 The category CG includes all the locally compact Hausdorff spaces and the spaces satisfying the first axiom of countabilaty (in particular, all metric spaces). Proof - Let X be a locally compact Hausdorff space. Given that U is an arbitrary subset of X and that 2 E 0, take a neighborhood V, of x whose closure is compact; then
v,
Now suppose that X satisfies the first axiom of countability. We take as a compact set C associated to x E U c X the set {xc) together with a sequence converging to 2. 0
Lemma B.3 Let f : X --+ Y be a function with X E Cg and Y E Top. Then f is continuous iff its restriction t o any compact subset of X is continuous. Proof - Let K be an arbitrary closed subset of Y ; then, for every compact subset C c X, f - ' ( K ) n C = f c ' ( K ) - where f c is the restriction of f to C - is closed by hypothesis. This shows that f is continuous. 0 Compactly generated spaces are indeed very much related to locally compact spaces, as we can see from the following result:
279 Theorem B.4 A Hausdorff space X is compactly generated iff it is the quotient of a locally compact Hausdorg space.
Proof -
j
Index all compact subsets of X in a set A and define the
surjection c:
u CA+X
XEA
to be the function whose restriction to any CX is the identity map. The topological sum C = UXEA CA is compactly generated and so, by Lemma B.3, c is continuous. Furthermore, C is locally compact. We need only prove that c is an identification map. In fact, if K c X is such that c - l ( K ) n CX is closed, for every A, then K is closed in X because c-l ( K )n CX= K n CA. + Suppose now that Y is locally compact Hausdorff and q : Y + X is an identification map. Let K c X be such that, for every X E A, K n Cx is closed in CA;we wish to prove that K is closed in X . Since Y is locally compact Hausdorff, every point y E Y has a neighborhood Vwwhose closure is compact; cover q - l ( K ) by these open sets:
q(c)
is closed in q(F)that is to say, Because q ( q ) is compact, K n there is a closed subset Kg of X such that
and consequently,
p ( K )n
= q-I(K,) n V ,
is closed in Y . This means that, for every compact subset C c Y , qW1(Kn ) C is closed in C. Since Y E CG, it follows that q - l ( K ) is closed in Y and therefore, K is closed. in X. 0
Corollary B.5 If 'I' E CG and q : Y + X is an identification map, then X E Cg.
280
APPENDIX B. C O M P A C T L Y GENERATED SPACES
Proof - By the theorem, there exists a locally compact Hausdorff space 2 and an identification map p : 2 -+ Y ; then the composition
is again an identification and once more by Theorem B.4, X is compactly generated. 0 The Cartesian product of two compactly generated spaces (with the product topology) is not necessarily a compactly generated space (there exists a celebrated example by C.H.Dowker in this respect - see [15,Example 2, Section 2.21). We circumvent this problem with the following stratagem. Firstly, denote by X x Y the usual Cartesian product; then, define the product X 8 Y in CG to be the space with the same underlying set as X x Y but with the compactly generated topology. Indeed, several times in the sequel we shall want to change the topology of a Hausdorff space X into the finer compactly generated topology; to indicate this change of topology in a more consistent manner, we shall denote the new topological space obtained from X with k ( X ) ; thus, with this convention, X 8 Y = k ( X x Y ) . At this point one should notice that for any Hausdorff space X , the new space k ( X )is also Hausdorff; moreover, the identity function 1s : k ( X ) 3 X is continuous.
Theorem B.6 If X is locally compact Hausdorff and Y E Cg, then
X @ Y Z X X Y . Proof - Let A c X x Y be such that, for every compact subset C C X x Y ,A n C is closed. We wish to prove that A is closed, i.e., that ( X x Y ) \ A is open. To this end, take arbitrarily a point (z,,yo) E ( X x Y ) \ A ; since X is locally compact Hausdorff, there exists an open set U of X containing zo and such that the closure g is compact. Since x {yo} is compact, A n ( g x {yo}) is closed. Hence there is an open set If c U such that 2, E V , is compact and
v
(V x
{yo})
nA
=8
(v
.
Let B be the projection into Y of A n x Y ) . If C c Y is compact, An(V x C) is closed in r/ x C and hence, is compact; but then, B n C is closed. Because Y E Cg, it follows that B is closed in Y . To complete
281 the proof, observe that (zo,y o ) belongs to the open set V x (Y \ B ) and that
V x (Y \ B )
c (X x Y ) \ A .
0
Observe that Dowker's example has consequences also for function spaces: in fact, if X consists of just two points and Y E Top, then M ( X , Y ) = Y x Y . Then, for every X , Y E CG, we take the function space of all maps from X into Y in CG to be
Y" = k ( M ( X ,Y ) ) (note that M ( X , Y ) is Hausdorff because Y is Hausdorff - see Exercise 1.1.2). Similarly, we define the function space of based maps from ( X , z o )E CG to ( Y , y o ) E CG to be
(Y,yo)(s'"o) = k ( M , ( X , Y ) ). We shall convene that from now on, whenever we talk about products or function spaces of compactly generated spaces, it is to those specific product or function spaces we defined above that we are referring to. The next theorem paraphrases Theorem 1.1.2; it is the so-called exponential law for CG.
Theorem B.7 For every X , Y,2 E CG, a function
as
continuous i f the function
Proof - As we have seen in Theorem 1.1.2, there are no restrictions on any of the spaces involved for the proof of the necessary condition. Thus, let us assume that is continuous. We first prove that, for every compact subset K of X, the restriction of f to K 8 2 is continuous. In fact, take the inclusion map i : K ~f X and its induced map i" : Y s --+ Y K and consider the commutative diagram of Figure B.1. The
f'
282
APPENDIX B. COMPACTLY GENERATED SPACES
Y FIGURE B . l evaluation function E is continuous because K is compact (see Theorem 1.1.1) and thus, f I K @ J Zis continuous. This shows that the restriction of f to any compact subset C c X @J 2 is continuous: in fact, the projection p q : X 8 2 3 X is continuous and C c p q ( C ) 8 2. Finally, f is continuous by Lemma B.3. 0
Corollary B.8 For every X , Y E Cg, the evaluation function
) f(z) is continuous. given by ~ ( z , f = Proof - The function
E:
is adjoint to the identity map
and so, is continuous by the sufficiency part of the previous theorem. 0
Lemma B.9 Let X , Y be given Hausdorfl spaces and let f : X + Y be a continuous function; let k( f) : k ( X ) k ( Y ) be the function which coincides with f at the set-theoretical level. Then k ( f ) is continuous. --f
Proof - It is enough to show that the restriction of k ( f ) to any compact subset C c k ( X ) is continuous. We begin by noticing that X and k ( X ) have the same compact subsets: If C c k ( X ) is compact, l.y(C) = C is compact in S ;now suppose that C c X is compact
283 and let C' be C with the relative topology from k ( X ) . The identity function 1~ : C' t C is continuous; we wish to prove that its inverse is also continuous and so, C' E C is compact in k ( X ) . Let K c C' be closed;, since K meets every compact subset of X in a closed set, K n C = K is closed in C proving thereby that C' + C is continuous. Now let C' c k ( X ) be compact. B y what we have just seen, the same set C with its topology in X is compact; since f is continuous, f (C) is compact in Y and so is f ( C ) ' in k ( Y ) .But then, the restriction k( f ) I C' is continuous as a composition of continuous functions, namely the identity maps lc, l f ( qand the map f I C. 0
Theorem B.10 For every X , Y , Z E CG, YxBz E (Y-')'. Proof - Let
B : M ( X @ Z , Y )+ M ( Z , M ( X , Y ) )
f.
be the function taking any map f : X @ Z -+ Y into its adjoint By Theorem B.7, 9 is a bijection. The sets W K ,of~ all maps f : X + Y such that f ( K ) C U ,for all K c X compact and U c Y open, form a sub-basis S for the open sets of M ( X , Y ) ;by the unbased version of Lemma 1.1.5, the sets &(L, W K , U ) of maps 9 :z
-
M ( X , Y ) 7 s(L)c w c u
c Z compact, form a sub-basis for the open sets of M(2, M ( X , Y ) ) . On the other hand, the sets
with L
W K ~= L (9 , ~E M ( X
2, Y ) I g ( K
L) c
u)
with K c X compact, L c 2 compact and U c Y open form a sub, K,~) basis for the open sets of M ( X @ 2,Y).Now, because g E F % ( L W iff 5 E WK@L,U, that
P ( F % ( L IT&[-) , = w/cc;;r.,c. ; Lemma B.9 then proves that
-
k ( 4 ) : y.v'&z is a homeomorphism. 0
1z
7.1-
(1
284
APPENDIX B. COMPACTLY GENERATED SPACES
EXERCISES B.l Let X and Y be given Hausdorff spaces. Prove that
k ( X ) @ k ( Y )2
qx x Y ) .
B.2 A subset A of X
E Top is said to be compactly closed if, for every compact Hausdorff space K and every map f : K -+ X , f - ' ( A ) is closed in K . A space X E Top is said to be compactly closed if every compactly closed subset of X is closed. Let CTop be the category of all compactly closed spaces. Prove that CG is a subcategory of C Top.
B.3 Let X and Y be compactly closed spaces. Prove that the space X @ Y , obtained by first taking the usual Cartesian product space X x Y and then refining its topology to the compactly closed topology, is a product in the category CTop.
B.4 A compactly closed space X is weak Hausdorflif the diagonal space A X is closed in X @ X . Prove that, X is weak Hausdorff iff for every compact Hausdorff space K and every map f : K + X , f ( K ) is closed in X.
B.5 Prove that the compactly closed weak Hausdorff spaces form a subcategory of CTop. Show that the former category has the following advantage over the latter: if ( X ,A ) is a pair of compactly closed weak Hausdorff spaces, so is X / A .