In algebraic topology, people work with topological spaces. This entails doing a lot of operations with topological spaces, i.e., taking products, quotients, pushout, limits, colimits, exponentials, etc. But the category \( \mathsf{Top} \) of topological spaces is not so good to work with. For example, we would want to construct the space of maps \( \mathsf{Top}(X, Y) \) and give a nice topology, so that the natural adjunction \[ \displaystyle \mathsf{Top}(X, \mathsf{Top}(Y, Z)) \cong \mathsf{Top}(X \times Y, Z) \] is a homeomorphism of spaces. Well, the best topology I can think of giving on the set \( \mathsf{Top}(X, Y) \) is the compact-open topology. And indeed there is a theorem of this sort in [Swi02].
Theorem 1 ([Swi02], Theorem 0.11). The natural map \[ \displaystyle \mathsf{Top}(X \times Y, Z) \rightarrow \mathsf{Top}(X, \mathsf{Top}(Y, Z)) \] is a homeomorphism if \( X \) and \( Y \) are Hausdorff and \( Y \) is locally compact.
I don’t know of a counterexample without the Hausdorff and locally compact hypotheses, but these conditions are probably crucial. If so, this is not very satisfying. Many spaces we work with are in fact Hausdorff, but infinite CW-complexes are not generally locally compact.
This calls for a construction of a "convenient category" to do algebraic topology. This term was used by Steenrod [Ste67], who proposed the category \( \mathsf{CG} \) of compactly generated (Hausdorff) spaces. But it turns out that a smaller category \( \mathsf{CGWH} \) is better behaved. This category is what we are after. For a comprehensive exposition, check out the draft of a book written by Strickland.
The whole material is a bit technical, but I think it is worth knowing how the category is constructed. Firstly because it is the groundwork for a lot of algebraic topology, and secondly because many of the ideas might be useful. Look how similar the definition of weak Hausdorff spaces is to the definition of separated schemes.
1. Compactly generated spaces#
Honestly I don’t know what motivated people to look at these spaces. It’s just astonishing that things work out nicely.
Definition 2. A subset \( A \subseteq X \) is called compactly closed if for any compact Hausdorff \( K \) and a continuous map \( f : K \rightarrow X \), the preimage \( f^{-1}(A) \subseteq K \) is closed.
Clearly closed sets are compactly closed.
Definition 3. For a space \( X \in \mathsf{Top} \), define a new topological space \( kX \) with the same underlying set, but so that a set \( A \subseteq kX \) is closed if and only if \( A \subseteq X \) is compactly closed.
Then \( kX \) has a finer topology than \( X \), i.e., has more closed/open sets. The natural "identity" map \( kX \rightarrow X \) is a continuous map.
Proposition 4. The \( k \)-ification \( k : \mathsf{Top} \rightarrow \mathsf{Top} \) is a functor.
Proof. Suppose \( f : X \rightarrow Y \) is any continuous map and \( A \subseteq Y \) is compactly closed. For any map \( g : K \rightarrow X \), the set \( g^{-1}(f^{-1}(A)) = (fg)^{-1}(A) \) is closed in \( K \). Thus \( f^{-1}(A) \) is compactly closed in \( X \). This means that \( f : kX \rightarrow kY \) is continuous. ▨
Proposition 5. For any \( X \in \mathsf{Top} \), the map \( k^2 X \rightarrow k X \) is a homeomorphism.
Proof. Given a compact Hausdorff space \( K \) and a (set) map \( f : K \rightarrow X \), the map \( f : K \rightarrow X \) is continuous if and only if \( f : K \rightarrow kX \) is continuous, trivially. So the compactly closed sets of \( X \) are the same as the compactly closed sets of \( kX \). In other words, \( k^2 X \rightarrow k X \) is a homeomorphism. ▨
Definition 6. A space \( X \in \mathsf{Top} \) is called compactly generated or a \( k \)-space if \( kX \rightarrow X \) is a homeomorphism. The full subcategory of \( \mathsf{Top} \) consisting of compactly generated spaces is denoted \( \mathsf{CG} \).
With this terminology, we are allowed to say that \( kX \) is compactly generated space for all \( X \in \mathsf{Top} \).
Proposition 7. The \( k \)-ification functor \( k : \mathsf{Top} \rightarrow \mathsf{CG} \) is right adjoint to the forgetful functor \( \mathsf{CG} \rightarrow \mathsf{Top} \).
Proof. For spaces \( X \in \mathsf{CG} \) and \( Y \in \mathsf{Top} \), it suffices to show that \( f : X \rightarrow Y \) is continuous if and only if \( f : X \rightarrow kY \) is continuous. Since \( kY \) is finer, we assume that \( X \rightarrow Y \) is continuous and show that \( X \rightarrow kY \) is continuous. But \( kX \rightarrow kY \) is continuous and \( kX \cong X \). ▨
This is somewhat unexpected, because normally forgetful functors are right adjoints.
Corollary 8. \( k \)-ification commutes with limits. In particular, the category \( \mathsf{CG} \) is complete because \( \mathsf{Top} \) is complete. Moreover, the underlying set of the limit is the limit of the underlying sets.
Lemma 9. Disjoint unions of compactly generated spaces are compactly generated. Quotients of compactly generated spaces by equivalence relations are compactly generated.
Proof. Take \( X _ i \in \mathsf{CG} \). Suppose that \( \coprod _ i A _ i \subseteq \coprod _ i X _ i \) is a compactly generated set. Then for any compact Hausdorff \( K \) and \( f : K \rightarrow X _ i \), we can compose it with \( \iota _ i : X _ i \hookrightarrow \coprod _ i X _ i \) to get a map \( \iota _ i \circ f : K \rightarrow \coprod _ i X _ i \). Clearly \( (\iota _ i \circ f)^{-1}(\coprod _ i A _ i) = f^{-1}(A _ i) \) and it is closed. This implies that \( A _ i \) is compactly closed, hence closed in \( X _ i \). Thus \( \coprod _ i A _ i \) is also closed in \( \coprod _ i X _ i \).
Take \( X \in \mathsf{CG} \) and a quotient map \( q : X \rightarrow {X/\sim} = Y \). The topology on \( Y \) is such that \( A \subseteq Y \) is closed if and only if \( q^{-1}(A) \) is closed in \( X \). If \( A \subseteq Y \) is compactly generated, for any \( f : K \rightarrow X \) the inverse image \( f^{-1}(q^{-1}(A)) = (q \circ f)^{-1}(A) \) is closed in \( K \). Then \( q^{-1}(A) \) is closed in \( X \) because \( X \in \mathsf{CG} \) and thus \( A \subseteq Y \) is closed. This shows that \( Y \in \mathsf{CG} \). ▨
Proposition 10. Colimits in \( \mathsf{CG} \) can simply be computed in \( \mathsf{Top} \). In particular, \( \mathsf{CG} \) is cocomplete because \( \mathsf{Top} \) is, and the underlying set of the colimit is the colimit of the underlying sets.
Proof. This is immediate because colimits in \( \mathsf{Top} \) can be constructed by taking disjoint unions and quotients. The colimit of compactly generated spaces in the category \( \mathsf{Top} \) is a compactly generated space. Thus it is also the colimit in \( \mathsf{CG} \). ▨
Notice that so far we have not used anything about "compact Hausdorff" in the definitions. I could have replaced them with "Lindelöf" and still gotten the same propositions. (I don’t even know what Lindelöf means.)
Example 1. Every locally compact Hausdorff space is in \( \mathsf{CG} \). For each point \( p \in X \), where \( X \) is locally compact, take a compact neighborhood, i.e., a neighborhood \( U _ p \subseteq X \) such that \( \overline{U _ p} \) is compact. Suppose that \( A \) is compactly open, and let us show that \( A \) is open. Take the inclusion map \( \overline{U _ p} \hookrightarrow X \), and we see that \( \overline{U _ p} \cap A \) is open in \( \overline{U _ p} \). Hence \( U _ p \cap A \) is open. Now \( A = \bigcup _ p (U _ p \cap A) \) is open.
Example 2. It follows that closed disks are in \( \mathsf{CG} \). Since we can take arbitrary colimits, all CW-complexes are also in \( \mathsf{CG} \).
Example 3. This example is taken from a blog post. Let \( X = \mathbb{R} \setminus \{1^{-1}, 2^{-1}, \ldots \} \) with the subspace topology and \( Y = \mathbb{R}/\mathbb{Z} \) be the countably infinite wedge of circles. I claim that \( X \times _ {\mathsf{Top}} Y \) is not in \( \mathsf{CG} \). Consider \[ \displaystyle A = \{ (1/i + a _ i/j, i + 1/2j) \in X \times Y : i, j \in \mathbb{N} \}, \quad a _ i = (1/i - 1/(i+1)) / 2. \] Its closure contains \( (0, 0) \) and thus \( A \) is not closed. But for any compact subset \( K \subseteq X \times Y \), the set \( K \cap A \) has only finitely many points. This is because for fixed \( i \), there are only finitely many \( j \), and also there can be only finitely many \( i \). Hence \( A \) is a compactly closed but not closed set.
It sometimes happens for CW-complexes \( X \) and \( Y \) that \( X \times _ {\mathsf{Top}} Y \) and \( k(X \times _ {\mathsf{Top}} Y) = X \times _ {\mathsf{CG}} Y \) are different. The \( \mathsf{CG} \) product is preferable, because \( X \times _ {\mathsf{CG}} Y \) is a CW-complex with (\( p+q \))-cells corresponding to products of \( p \)-cells and \( q \)-cells.
2. Mapping spaces#
We now define the topology on the space of continuous maps.
Definition 11. For \( X, Y \in \mathsf{CG} \), consider the space \( C _ 0(X, Y) \) of continuous maps \( X \rightarrow Y \), equipped with the compact-open topology generated by the sets \[ \displaystyle W(u, K, U) = \{ f : X \rightarrow Y \text{ with } f(u(K)) \subseteq U \}, \] where \( K \) is compact Hausdorff and \( u : K \rightarrow X \). (This is slightly different from the standard definition, but we use this because we have set "compact Hausdorff" as the base for our discussion.) We further define \( C(X,Y) = k C _ 0(X,Y) \).
Proposition 12. For any \( X, Y \in \mathsf{CG} \), \( C _ 0(X, -) \) is a contravariant functor and \( C _ 0(-, Y) \) is a covariant functor. Consequently, \( C(X, -) \) and \( C(-, Y) \) are also functors.
Proof. We want to show that if \( f : X \rightarrow Z \) is continuous then \( f^\ast : C _ 0(Z, Y) \rightarrow C _ 0(X, Y) \) is continuous. It suffices to show that \( (f^\ast)^{-1}(W(u, K, U)) = W(f \circ u, K, U) \) is open, but this is by definition.
Next, we want to show that if \( g : Y \rightarrow Z \) then \( g _ \ast : C _ 0(X, Y) \rightarrow C _ 0(X, Z) \) is continuous. This time, \( (g _ \ast)^{-1}(W(u, K, U)) = W(u, K, g^{-1}(U)) \) is open. ▨
We finally see why we have chosen "compact Hausdorff" in our definitions.
Proposition 13. For \( X, Y \in \mathsf{CG} \), the evaluation map \( \mathrm{ev} _ {X,Y} : X \times _ {\mathsf{CG}} C(X, Y) \rightarrow Y \) and the injection map \( \mathrm{inj} _ {X,Y} : Y \rightarrow C(X, X \times _ {\mathsf{CG}} Y) \) are continuous.
Proof. Let’s look at the evaluation map. Take an open set \( U \subseteq Y \). It suffices to show that for any \( u : K \rightarrow X \times _ {\mathsf{Top}} C _ 0(X, Y) \), the inverse image \( V = (\mathrm{ev} \circ u)^{-1}(U) \) is open in \( K \). The map \( u \) contains two data: \( u _ 1 : K \rightarrow X \) and \( u _ 2 : K \rightarrow C _ 0(X, Y) \). Then we can write \[ \displaystyle V = \{ a \in K : u _ 2(a)(u _ 1(a)) \in U \}. \] Fix any \( a \in V \). Since \( u _ 2(a) \circ u _ 1 : K \rightarrow Y \) is continuous, we find a compact neighborhood \( L \subseteq K \) of \( a \) such that \( u _ 2(a)(u _ 1(L)) \subseteq U \). (Every compact Hausdorff space is regular.) Then \( u _ 2(a) \in W(u _ 1, L, U) \) so write \( N = u _ 2^{-1}(W(u _ 1, L, U)) \). For every \( b \in L \cap N \), \[ \displaystyle u _ 2(b)(u _ 1(b)) \in u _ 2(b)(u _ 1(L)) \subseteq U. \] Since \( L \cap N \) contains an open neighborhood of \( a \in K \), it follows that \( V \) is open.
We now look at the injection map. It suffices to show that \( \mathrm{inj} _ {X,Y} : Y \rightarrow C _ 0(X, X \times _ {\mathsf{CG}} Y) \) is continuous, since we can apply \( k \) afterwards. We show that the set \( \mathrm{inj} _ {X,Y}^{-1}( W(u, K, U)) \subseteq Y \) is open for any \( u : K \rightarrow X \) and \( U \subseteq X \times _ {\mathsf{CG}} Y \). Since \( Y \in \mathsf{CG} \), we show that \[ \displaystyle v^{-1}(\mathrm{inj} _ {X,Y}^{-1}(W(u,K,U))) = \{ b \in L : (u \times v)(K \times \{b\}) \subseteq U \} \] is open in \( L \), for every \( v : L \rightarrow Y \). But this is true by the "tube lemma", because \( K \) and \( L \) are compact. ▨
Theorem 14. For \( X, Y, Z \in \mathsf{CG} \), there is a natural adjoint map \( \mathrm{adj} _ {X,Y,Z} : C(X, C(Y, Z)) \rightarrow C(X \times _ {\mathsf{CG}} Y, Z) \) which is a homeomorphism.
Proof. We first show that it is a bijection at the level of sets. If \( f : X \rightarrow C(Y, Z) \) is continuous, its image \[ \displaystyle X \times _ {\mathsf{CG}} Y \xrightarrow{f \times \mathrm{id}} C(Y, Z) \times _ {\mathsf{CG}} Y \xrightarrow{\mathrm{ev}} Z \] is continuous. On the other hand, if \( g : X \times _ {\mathsf{CG}} Y \rightarrow Z \) is continuous, \[ \displaystyle X \xrightarrow{\mathrm{inj}} C(Y, X \times _ {\mathsf{CG}} Y) \xrightarrow{g _ \ast} C(Y, Z) \] is continuous. This shows that \( \mathrm{adj} _ {X,Y,Z} \) is a bijection.
Now for each \( W \in \mathsf{CG} \), we have natural bijections \[ \displaystyle \begin{aligned} C(W,C(X,C(Y,Z))) & \displaystyle\rightarrow C(W \times X, C(Y,Z)) \\ & \displaystyle\rightarrow C(W \times X \times Y, Z) \leftarrow C(W, C(X \times Y, Z)). \end{aligned} \] By Yoneda, it follows that \( C(X, C(Y, Z)) \cong C(X \times _ {\mathsf{CG}} Y, Z) \) as spaces. ▨
Thus we have obtained a category \( \mathsf{CG} \), which:
- is complete and cocomplete,
- contains all CW-complexes,
- and has a Cartesian closed monoidal structure.
Here’s another fun fact.
Proposition 15. For \( X, Y, Z \in \mathsf{CG} \), the composition map \( C(X, Y) \times _ {\mathsf{CG}} C(Y, Z) \rightarrow C(X, Z) \) is continuous.
Proof. The adjoint of the map \[ \displaystyle X \times C(X, Y) \times C(Y, Z) \xrightarrow{\mathrm{ev} \times \mathrm{id}} Y \times C(Y, Z) \xrightarrow{\mathrm{ev}} Z \] is the desired map. ▨
3. Weakly Hausdorff spaces#
We now have a pretty good category. But this category still contains some bad topological spaces, like \( \mathrm{Spec} k[x] _ {(x)} \). (This is the space \( \{a, b\} \) with open sets \( \emptyset, \{a\}, \{a, b\} \).) These are not of interest in algebraic topology, and we would like to remove them.
Proposition 16. For a space \( X \in \mathsf{CG} \), the following are equivalent:
- (a) For every \( u : K \rightarrow X \), its image \( u(K) \) is closed in \( X \).
- (b) The diagonal map \( \Delta : X \rightarrow X \times _ {\mathsf{CG}} X \) is a closed inclusion, i.e., a homeomorphism to a closed subset.
If \( X \in \mathsf{CG} \) satisfies these properties, we say that \( X \) is compactly generated weakly Hausdorff. We also write the full subcategory of these spaces as \( \mathsf{CGWH} \).
Proof. (a) \( \Rightarrow \) (b) It is not hard to verify that \( X \rightarrow X \times _ {\mathsf{CG}} X \) is always an inclusion map. What we want to check is that the image \( \Delta _ X \subseteq X \times _ {\mathsf{CG}} X \) is closed. That is, we want to show that for any map \( u = (u _ 1, u _ 2) : K \rightarrow X \times _ {\mathsf{Top}} X \), the inverse image \[ \displaystyle u^{-1}(\Delta _ X) = \{ a \in K : u _ 1(a) = u _ 2(a) \} \] is closed.
Pick a point \( a \in K \) with \( u _ 1(a) \neq u _ 2(a) \). Since \( \{a\} \) is compact in \( K \), \( \{u _ 2(a)\} \) is closed in \( X \). Then \( \{ b \in K : u _ 1(b) \neq u _ 2(a) \} \) is open in \( K \). By regularity, there is an open neighborhood \( a \in U \) such that \( b \in \overline{U} \) implies \( u _ 1(b) \neq u _ 2(a) \). Again, \( \overline{U} \) is compact and hence \( u _ 2(\overline{U}) \) is closed in \( X \). By the same argument one finds an open neighborhood \( a \in V \) such that \( b \in \overline{V} \) implies \( u _ 2(c) \notin u _ 1(\overline{U}) \). If \( b \in U \cap V \) then \( u _ 2(b) \notin u _ 1(\overline{U}) \) but \( u _ 1(b) \in u _ 1(\overline{U}) \). Hence \( U \cap V \) is a neighborhood of \( a \) contained in \( u^{-1}(\Delta _ X) \).
(b) \( \Rightarrow \) (a) For any \( u : K \rightarrow X \) and \( v : L \rightarrow X \), we have \[ \displaystyle v^{-1}(u(K)) = \pi _ L(\{ (a,b) \in K \times L : u(a) = v(b) \}) = \pi _ L((u \times v)^{-1}(\Delta _ X)). \] But \( \pi _ L : K \times L \rightarrow L \) is a closed map since \( K \) is compact, and hence \( v^{-1}(u(K)) \) is always closed. This means that \( u(K) \) is always compactly closed and hence closed. ▨
It is clear that weakly Hausdorff is between Hausdorff and T1(points are closed).
Remark 1. If we work in the category \( \mathsf{Top} \), (b) is exactly the Hausdorff condition. More precisely, \( \Delta : X \rightarrow X \times _ {\mathsf{Top}} X \) is a closed inclusion if and only if \( X \) is Hausdorff.
Like in the case of passing from \( \mathsf{Top} \) to \( \mathsf{CG} \), there is a forgetful functor \( \mathsf{CGWH} \rightarrow \mathsf{CG} \), and it has an adjoint. But before we construct this functor, we need some facts about taking quotients in \( \mathsf{CG} \). Recall that we have proven that quotients of \( \mathsf{CG} \) spaces by equivalence relations are always \( \mathsf{CG} \), Lemma 9.
Proposition 17. For \( X, Y \in \mathsf{CG} \) and \( E \) an equivalence relation on \( X \), there is a natural homeomorphism \( (X \times _ {\mathsf{CG}} Y) / (E \times \mathrm{id}) \cong (X/E) \times _ {\mathsf{CG}} Y \).
Proof. Let us write \( Q = (X \times _ {\mathsf{CG}} Y) / (E \times \mathrm{id}) \). First the map \( X \times _ {\mathsf{CG}} Y \rightarrow (X/E) \times _ {\mathsf{CG}} Y \) respects the relation \( E \times \mathrm{id} \) and thus factors through the quotient. Thus we have a map \( Q \rightarrow (X/E) \times _ {\mathsf{CG}} Y \).
For the other map, we look at the projection map \( X \times _ {\mathsf{CG}} Y \rightarrow Q \) and curry it to get a map \( X \rightarrow C(Y, Q) \). This respects \( E \) on the level of sets, and so factors to give a map \( X/E \rightarrow C(Y, Q) \). Then curry again to get a map \( (X/E) \times _ {\mathsf{CG}} Y \rightarrow Q \). These two are clearly inverses. ▨
Lemma 18. For an equivalence relation \( E \) on \( X \in \mathsf{CG} \), the set \( E \subseteq X \times _ {\mathsf{CG}} X \) is closed if and only if \( X / E \in \mathsf{CGWH} \).
Proof. We have \( (X \times _ {\mathsf{CG}} X) / (E \times E) \cong (X/E) \times _ {\mathsf{CG}} (X/E) \) by applying the previous proposition twice. Thus \( \Delta _ {X/E} \) is closed in \( (X/E) \times _ {\mathsf{CG}} (X/E) \) if and only if \( q^{-1}(\Delta _ {X/E}) \subseteq X \times _ {\mathsf{CG}} X \) is closed, where \( q : X \times _ {\mathsf{CG}} X \rightarrow (X/E) \times _ {\mathsf{CG}} (X/E) \) is the projection map. But \( q^{-1}(\Delta _ {X/E}) = E \). ▨
Theorem 19. There is a functor \( h : \mathsf{CG} \rightarrow \mathsf{CGWH} \) that is a left adjoint to the forgetful functor \( \mathsf{CGWH} \rightarrow \mathsf{CG} \).
Proof. We explicitly construct this functor in the following way. For \( X \in \mathsf{CG} \), consider the smallest equivalent relation \( E \subseteq X \times _ {\mathsf{CG}} X \) that is closed. (You can just take the intersection of all closed equivalence relations.) Then we know that \( hX = X/E \in \mathsf{CGWH} \). There is of course a natural (projection) map \( X \rightarrow hX \).
To show that this is left adjoint to the forgetful functor, take \( X \in \mathsf{CG} \) and \( Y \in \mathsf{CGWH} \). We want to show that every \( f : X \rightarrow Y \) factors through \( X \rightarrow hX \). Since \( Y \in \mathsf{CGWH} \), \( \Delta _ Y \subseteq Y \times _ {\mathsf{CG}} Y \) is closed and hence \( (f \times f)^{-1}(\Delta _ Y) \) is also closed in \( X \times _ {\mathsf{CG}} Y \). This is an equivalence relation that contains \( E \), since it is closed. Thus \( X \rightarrow Y \) respects \( E \) and factors as \( X \rightarrow X/E = hX \rightarrow Y \).
It follows that \( h \) is actually a functor. For \( f : X \rightarrow Y \) where \( X, Y \in \mathsf{CG} \), we have \( X \rightarrow Y \rightarrow hY \), which in turn gives \( hX \rightarrow hY \). ▨
Corollary 20. The functor \( h \) commutes with colimits. In particular, the category \( \mathsf{CGHW} \) is cocomplete because \( \mathsf{CG} \) is.
One thing noteworthy is that the underlying set of the colimit can now be different from the colimit of the underlying sets.
Example 4. Consider the pushout \( \ast \leftarrow [0, 1) \rightarrow [0,1] \). In \( \mathsf{Top} \) or \( \mathsf{CG} \), it is going to be \( \{a,b\} \) with \( \{a\} \) open but \( \{b\} \) not open. (Or \( \mathrm{Spec} k[x] _ {(x)} \).) Applying \( h \) gives a point.
Proposition 21. Small limits in \( \mathsf{CGWH} \) can be computed in \( \mathsf{CG} \). In other words, if \( X _ i \in \mathsf{CGWH} \) then \( \varprojlim _ {\mathsf{CG}} X _ i \in \mathsf{CGWH} \). Hence \( \mathsf{CGWH} \) is complete and the underlying set of the limit is the limit of the underlying sets.
Proof. Let us write \( L = \varinjlim _ {\mathsf{CG}} X _ i \). We are trying to show that \( \Delta _ L \subseteq L \times _ {\mathsf{CG}} L \) is closed. Note that there are natural maps \( p _ i : L \times _ {\mathsf{CG}} L \rightarrow X _ i \times _ {\mathsf{CG}} X _ i \). Since \( L \) as a set is the limit of \( X _ i \) as sets, we easily see that \[ \displaystyle \Delta _ L = \bigcap _ {i} p _ i^{-1}(\Delta _ {X _ i}). \] Thus \( \Delta _ L \) is a intersection of closed sets, hence closed. ▨
So the smaller category \( \mathsf{CGWH} \) is also complete and cocomplete. What about being Cartesian closed?
Proposition 22. If \( X \in \mathsf{CG} \) and \( Y \in \mathsf{CGWH} \) then \( C(X, Y) \in \mathsf{CGWH} \). Thus \( \mathsf{CGWH} \) is Cartesian closed.
Proof. Define \( \mathrm{ev} _ x : C(X,Y) \cong \{x\} \times C(X,Y) \rightarrow X \times C(X,Y) \xrightarrow{\mathrm{ev}} Y \). Then the diagonal \( \Delta _ {C(X,Y)} \) can be written as \[ \displaystyle \Delta _ {C(X,Y)} = \bigcap _ {x \in X} (\mathrm{ev} _ x \times \mathrm{ev} _ x)^{-1} (\Delta _ Y) \] which is closed. ▨
Thus we have constructed the category \( \mathsf{CGWH} \) that:
- contains locally compact Hausdorff spaces,
- contains a locally compact space if and only if it is Hausdorff,
- is complete and cocomplete,
- thus contains all CW-complexes,
- preserves limits under the forgetful functor to sets,
- has a Cartesian closed monoidal structure.
References#
[Ste67] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. MR 0210075
[Swi02] Robert M. Switzer, Algebraic topology—homotopy and homology, Classics in Mathematics, Springer-Verlag, Berlin, 2002, Reprint of the 1975 original [Springer, New York; MR0385836 (52 #6695)]. MR 1886843