COSTO TOTAL US$ 7,500
B.3. La Fibra Óptica
We briefly describe the category of spaces in which algebraic topologists cus- tomarily work. The ordinary category of spaces allows pathology that obstructs a clean development of the foundations. The homotopy and homology groups of spaces are supported on compact subspaces, and it turns out that if one assumes a separation property that is a little weaker than the Hausdorff property, then one can refine the point-set topology of spaces to eliminate such pathology without changing these invariants. We shall leave the proofs to the reader, but the wise reader will simply take our word for it, at least on a first reading: we do not want to overemphasize this material, the importance of which can only become apparent in retrospect.
1. The definition of compactly generated spaces
We shall understand compact spaces to be both compact and Hausdorff, fol- lowing Bourbaki. A spaceX is said to be “weak Hausdorff” ifg(K) is closed inX
for every mapg:K−→X from a compact spaceK intoX. When this holds, the image g(K) is Hausdorff and is therefore a compact subspace ofX. This separa- tion property lies betweenT1(points are closed) and Hausdorff, but it is not much weaker than the latter.
A subspaceAofXis said to be “compactly closed” ifg−1(A) is closed inKfor any mapg:K−→X from a compact spaceKintoX. WhenX is weak Hausdorff, this holds if and only if the intersection of A with each compact subset of X is closed. A spaceX is a “k-space” if every compactly closed subspace is closed.
A space X is “compactly generated” if it is a weak Hausdorff k-space. For example, any locally compact space and any weak Hausdorff space that satisfies the first axiom of countability (every point has a countable neighborhood basis) is compactly generated. We have expressed the definition in a form that should make the following statement clear.
Lemma. IfX is a compactly generated space andY is any space, then a func-
tionf :X −→Y is continuous if and only if its restriction to each compact subspace
K of X is continuous.
We can make a spaceX into ak-space by giving it a new topology in which a space is closed if and only if it is compactly closed in the original topology. We call the resulting space kX. Clearly the identity functionkX −→X is continuous. If
X is weak Hausdorff, then so iskX, hencekX is compactly generated. Moreover,
X andkXthen have exactly the same compact subsets.
WriteX×cY for the product ofX and Y with its usual topology and write
X×Y =k(X×cY). IfX andY are weak Hausdorff, thenX×Y =kX×kY. If
X is locally compact andY is compactly generated, thenX×Y =X×cY. 39
40 COMPACTLY GENERATED SPACES
By definition, a spaceX is Hausdorff if the diagonal subspace ∆X ={(x, x)}is closed inX×cX. The weak Hausdorff property admits a similar characterization. Lemma. If X is a k-space, then X is weak Hausdorff if and only if ∆X is
closed in X×X.
2. The category of compactly generated spaces
One major source of point-set level pathology can be passage to quotient spaces. Use of compactly generated topologies alleviates this.
Proposition. IfX is compactly generated andπ:X−→Y is a quotient map,
thenY is compactly generated if and only if(π×π)−1(∆Y) is closed inX×X.
The interpretation is that a quotient space of a compactly generated space by a “closed equivalence relation” is compactly generated. We are particularly interested in the following consequence.
Proposition. If X andY are compactly generated spaces, A is a closed sub-
space ofX, and f :A−→Y is any continuous map, then the pushout Y ∪fX is
compactly generated.
Another source of pathology is passage to colimits over sequences of maps
Xi −→Xi+1. When the given maps are inclusions, the colimit is the union of the setsXiwith the “topology of the union;” a set is closed if and only if its intersection with eachXi is closed.
Proposition. If {Xi} is a sequence of compactly generated spaces and inclu-
sionsXi−→Xi+1 with closed images, then colimXi is compactly generated.
We now adopt a more categorical point of view. We redefine U to be the
category of compactly generated spaces and continuous maps, and we redefineT
to be its subcategory of based spaces and based maps.
Let wU be the category of weak Hausdorff spaces. We have the functor k : wU −→U, and we have the forgetful functorj:U −→wU, which embedsU as
a full subcategory ofwU. Clearly
U(X, kY)∼=wU(jX, Y)
for X ∈ U and Y ∈ wU since the identity map kY −→ Y is continuous and
continuity of maps defined on compactly generated spaces is compactly determined. Thuskis right adjoint toj.
We can construct colimits and limits of spaces by performing these construc- tions on sets: they inherit topologies that give them the universal properties of colimits and limits in the classical category of spaces. Limits of weak Hausdorff spaces are weak Hausdorff, but limits of k-spaces need not be k-spaces. We con- struct limits of compactly generated spaces by applying the functorkto their limits as spaces. It is a categorical fact that functors which are right adjoints preserve limits, so this does give categorical limits inU. This is how we definedX×Y, for
example.
Point-set level colimits of weak Hausdorff spaces need not be weak Hausdorff. However, if a point-set level colimit of compactly generated spaces is weak Haus- dorff, then it is a k-space and therefore compactly generated. We shall only be interested in colimits in those cases where this holds. The propositions above give examples. In such cases, these constructions give categorical colimits inU.
2. THE CATEGORY OF COMPACTLY GENERATED SPACES 41
From here on, we agree that all given spaces are to be compactly generated, and we agree to redefine any construction on spaces by applying the functor k to it. For example, for spaces X and Y in U, we understand the function space
Map(X, Y) = YX to mean the set of continuous maps from X to Y with the k- ification of the standard compact-open topology; the latter topology has as basis the finite intersections of the subsets of the form{f|f(K)⊂U} for some compact subset K of X and open subset U of Y. This leads to the following adjointness homeomorphism, which holds without restriction when we work in the category of compactly generated spaces.
Proposition. For spaces X,Y, andZ inU, the canonical bijection Z(X×Y)∼= (ZY)X
is a homeomorphism.
Observe in particular that a homotopyX×I−→Y can equally well be viewed as a mapX −→YI. These adjoint, or “dual,” points of view will play an important role in the next two chapters.
PROBLEMS
(1) (a) Any subspace of a weak Hausdorff space is weak Hausdorff. (b) Any closed subspace of ak-space is ak-space.
(c) An open subsetU of a compactly generated space X is compactly generated if each point has an open neighborhood inX with closure contained in U.
(2) * A Tychonoff (or completely regular) spaceX is aT1-space (points are closed) such that for each pointx∈X and each closed subsetAsuch that
x /∈A, there is a functionf :X−→I such thatf(x) = 0 andf(a) = 1 if
a∈A. Prove the following (e.g., Kelley,General Topology).
(a) A space is Tychonoff if and only if it can be embedded in a cube (a product of copies of I).
(b) There are Tychonoff spaces that are not k-spaces, but every cube is a compact Hausdorff space.
(3) Brief essay: In view of Problems 1 and 2, what should we mean by a “subspace” of a compactly generated space. (We donot want to restrict the allowable set of subsets.)
CHAPTER 6