5. A Convenient Category of Spaces

In homotopy theory, it would be convenient to work with a category of spaces which has all limits, colimits, and enjoys nice properties about mapping space (especially the Exponential Law). The full category $\mathsf{Top}$ does not work since the Exponential Law fails. The subcategory of locally compact Hausdorff spaces has the Exponential Law, but does not preserve limits and colimits in general. It turns out that there is some complete and cocomplete category that sits in between locally compact Hausdorff spaces and all topological spaces, and enjoys the Exponential Law. Compactly generated weak Hausdorff spaces give such a category $\mathsf{CGWH}$, which we briefly discuss in this section. This will be a convenient category for homotopy theory.

Compactly generated space

Definition 5.1. A subset $Y\subset X$ is called "compactly closed" (or "k-closed") if $f^{-1}(Y)$ is closed in $K$ for every continuous map $f:K\to X$ with $K$ compact Hausdorff. We define a new topology on $X$, denoted by $kX$, where close subsets of $kX$ are compactly closed subsets of $X$. The identity$$kX\to X$$is a continuous map. $X$ is called compactly generated if $kX=X$.

Let $\mathsf{CG}$ denote the full subcategory of $\mathsf{Top}$ consisting of compactly generated spaces.

If a space $X$ is compactly generated, then for any $Y$, a map $f:X\to Y$ is continuous if and only if the composition $K\to X\to Y$ is continuous for any continuous $K\to X$ with $K$ compact Hausdorff. Note that$$k^{2}X=kX.$$

Proposition 5.2. Every locally compact Hausdorff space is compactly generated.

Proof. Let $X$ be locally compact Hausdorff and $Z$ be a k-closed subset. We need to show that $Z=\bar{Z}$ is closed.

Let $x\in \bar{Z}$. Since $X$ is locally compact Hausdorff, $x$ has a neighborhood $U$ with $K=\bar{U}$ compact Hausdorff. Then $x\in \overline{K\cap Z}$. Since $Z$ is k-closed, $K\cap Z$ is closed in $K$, hence closed in $X$. So $x\in Z$.

$\square$

Proposition 5.3. The assignment $X\to kX$ defines a functor $\mathsf{Top}\to \mathsf{CG}$, which is right adjoint to the embedding $i:\mathsf{CG}\subset \mathsf{Top}$. In other words, we have an adjoint pair

Proof. Let $X\in \mathsf{CG},Y\in \mathsf{Top}$, we need to show that $f:X\to Y$ is continuous if and only if the same map $f:X\to kY$ is continuous. Assume $f:X\to kY$ is continuous. Then the composition $X\to kY\to Y$ is continuous. Conversely, assume $f:X\to Y$ is continuous. Let $Z\subset Y$ be a k-closed subset. Then for any $g:K\to X$ with $K$ compact Hausdorff,$$g^{-1}(f^{-1}(Z))=(f\circ g{)}^{-1}(Z)$$is closed in $K$. It follows that $f^{-1}(Z)$ is k-closed in $X$, hence closed. So $f:X\to kY$ is continuous.

$\square$

Proposition 5.4. Let $X\in \mathsf{CG}$ and $p:X\to Y$ be a quotient map. Then $Y\in \mathsf{CG}$.

Proof. By Proposition 5.3, $p$ factor through $X\to kY$. Since the quotient topology is the finest topology making the quotient map continuous, we find $Y=kY$.

$\square$

Theorem 5.5. The category $\mathsf{CG}$ is complete and cocomplete. Colimits in $\mathsf{CG}$ inherit the colimits in $\mathsf{Top}$. The limits in $\mathsf{CG}$ are obtained by applying $k$ to the limits in $\mathsf{Top}$.

Proof. Let $F\in \mathsf{Fun}(I,\mathsf{CG})$ and $\hat{F}=i\circ F\in \mathsf{Fun}(I,\mathsf{Top})$, where $i:\mathsf{CG}\to \mathsf{Top}$ is the embedding.

Note that the left adjoint functor $i:\mathsf{CG}\to \mathsf{Top}$ preserves colimits. Since $F(i)\in \mathsf{CG}$, their coproduct ${\coprod }_{i\in I}F(i)$ in $\mathsf{Top}$ (given by the disjoint union) is in $\mathsf{CG}$. Since $\mathbf{colim}\hat{F}$ is a quotient of ${\coprod }_{i\in I}F(i)$, it also lies in $\mathsf{CG}$ by Proposition 5.4. This implies the statement about $\mathbf{colim}F$.

As the right adjoint functor $k:\mathsf{Top}\to \mathsf{CG}$ preserves limits,$$\mathbf{lim}F=\mathbf{lim}(k\circ \hat{F})=k\mathbf{lim}\hat{F}.$$

$\square$

Corollary 5.6. Let $\{X_i{\}}_{i\in I}$ be a family of objects in $\mathsf{CG}$. Then their product in $\mathsf{CG}$ is$$k\prod _{i\in I}{X}_i$$where $\prod _{i\in I}{X}_i$ is the topological product of $X_i$’s.

Definition 5.7. $$\boxed{\text{We will use ×,∏ to denote the product in CG and ×t,∏t to denote the product in Top.}}$$

Another corollary of Theorem 5.5 is the following.

Proposition 5.8. Assume $X$ is compactly generated and $Y$ is locally compact Hausdorff, then $X\times Y=X\stackrel{t}{\times }Y$.

Definition 5.9. Let $X,Y\in \mathsf{CG}$. We define the compactly generated topology on ${\mathrm{Hom}}_{\mathsf{Top}}(X,Y)$ by$$\mathrm{Map}(X,Y)=kC(X,Y)\in \mathsf{CG}.$$Here $C(X,Y)$ is the compact-open topology generated by$$\{f\in {\mathrm{Hom}}_{\mathsf{Top}}(X,Y)|f(g(K))\subset U,\text{for some g:K→X with K compact Hausdorff and U⊂Y open}\}.$$

Note that the compact-open topology we use here for $\mathsf{CG}$ is slightly different from the usual one: we ask for a map from $K$ which is compact Hausdorff. We will also use the exponential notation$$Y^X:=\mathrm{Map}(X,Y).$$

Lemma 5.10. Let $X,Y\in \mathsf{CG}$, $K$ compact Hausdorff and $f:K\to X$ continuous. Then the evaluation map$$\begin{array}{rcl}e{v}_K:\mathrm{Map}(X,Y)\stackrel{t}{\times }K& \to & Y,\\ (g,p)& \to & g(f(p))\end{array}$$is continuous. In particular, $\mathrm{Map}(X,Y)\times K\to Y$ is continuous.

Proof. Let $U\subset Y$ be open, and $(g,p)\in e{v}_K^{-1}(U)$. Then $g\circ f^{-1}(U)$ is open in $K$ and contains $p$. Since $K$ is compact Hausdorff, $p$ has a neighborhood $V$ such that $\bar{V}\subset g\circ f^{-1}(U)$. Then$$\{h|h(f(\bar{V}))\subset U\}\times V$$is an open neighborhood of $(g,p)$.

$\square$

Proposition 5.11. Let $X,Y\in \mathsf{CG}$. Then the evaluation map $\mathrm{Map}(X,Y)\times X\to Y$ is continuous.

Proof. Let $K$ be compact Hausdorff with a continuous map $K\to \mathrm{Map}(X,Y)\times X$. We need to show the composition $K\to \mathrm{Map}(X,Y)\times X\to Y$ is continuous. But this is the same as the composition$$K\to \mathrm{Map}(X,Y)\times K\to Y,$$which is continuous by the previous lemma.

$\square$

Proposition 5.12. Let $X,Y,Z\in \mathsf{CG}$ and $f:X\times Y\to Z$ continous. Then the induced map$$\begin{array}{rcl}\hat{f}:X& \to & \mathrm{Map}(Y,Z),\\ x& \mapsto & f(x,-)\end{array}$$is also continuous.

Proof. We need to show $\hat{f}:X\to C(Y,Z)$ is continuous. Let $h:K\to Y$ be a continuous map with $K$ compact Hausdorff, and $U\subset Z$ open. Let$$W=\{g:Y\to Z|g(h(K))\subset U\}.$$Let $x\in {\hat{f}}^{-1}(W)$, i.e., $f(x,h(K))\subset U$. Since $f$ is continuous and $K$ is compact, there exists an open neighborhood $V$ of $x$ such that $f(V,h(K))\subset U$. Then $V\subset f^{-1}(W)$ as required.

$\square$

Theorem 5.13 (Exponential Law). Let $X,Y,Z\in \mathsf{CG}$. Then the natural map$$\begin{array}{rcl}\mathrm{Map}(X\times Y,Z)& \to & \mathrm{Map}(X,\mathrm{Map}(Y,Z)),\\ f& \mapsto & f(x,-)\end{array}$$is a homeomorphism.

Proof. We first show that$${\mathrm{Hom}}_{\mathsf{Top}}(X\times Y,Z)\to {\mathrm{Hom}}_{\mathsf{Top}}(X,\mathrm{Map}(Y,Z))$$is a set isomorphism. Note that this map is well-defined by Proposition 5.12, which is obviously injective.

For any continuous $g:X\to \mathrm{Map}(Y,Z)$, we obtain$$f:X\times Y\stackrel{g\times 1}{\to }\mathrm{Map}(Y,Z)\times Y\to Z$$which is continuous. This proves the surjectivity and we have established the set isomorphism.

The fact on homeomorphism is a formal consequence. In fact, for any $W\in \mathsf{CG}$, we have$$\begin{array}{rl}{\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(X\times Y,Z))& \cong {\mathrm{Hom}}_{\mathsf{Top}}(W\times X\times Y,Z)\\ & \cong {\mathrm{Hom}}_{\mathsf{Top}}(W\times X,\mathrm{Map}(Y,Z))\\ & \cong {\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(X,\mathrm{Map}(Y,Z))).\end{array}$$This says that we have a natural isomorphism between the two functors$${\mathrm{Hom}}_{\mathsf{Top}}(-,\mathrm{Map}(X\times Y,Z))\cong {\mathrm{Hom}}_{\mathsf{Top}}(-,\mathrm{Map}(X,\mathrm{Map}(Y,Z))):\mathsf{CG}\to \mathsf{Set}.$$Then Yoneda Lemma gives rise to the homeomorphism$$\mathrm{Map}(X\times Y,Z)\cong \mathrm{Map}(X,\mathrm{Map}(Y,Z)).$$$\square$

Proposition 5.14. Let $X,Y,Z\in \mathsf{CG}$. Then the composition$$\begin{array}{rcl}\mathrm{Map}(X,Y)\times \mathrm{Map}(Y,Z)& \to & \mathrm{Map}(X,Z),\\ (f,g)& \mapsto & g\circ f\end{array}$$is continuous, i.e., a morphism in $\mathsf{CG}$.

Proof. This follows from the Exponential Law. By Yoneda Lemma, we need to find a natural transformation$${\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(X,Y)\times \mathrm{Map}(Y,Z))\to {\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(X,Z)),\forall W\in \mathsf{CG}.$$First we observe that$$\begin{array}{rl}{\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(X,Y)\times \mathrm{Map}(Y,Z))\cong & {\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(X,Y))\times {\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(Y,Z))\\ \cong & {\mathrm{Hom}}_{\mathsf{Top}}(W\times X,Y)\times {\mathrm{Hom}}_{\mathsf{Top}}(W\times Y,Z).\end{array}$$Now given two maps $f:W\times X\to Y,g:W\times Y\to Z$, we consider the composition$$W\times X\stackrel{\Delta \times 1_X}{\to }W\times W\times X\stackrel{1\times f}{\to }W\times Y\stackrel{g}{\to }Z.$$Here $\Delta :W\to W\times W$ is the diagonal map. This gives naturally the required element of$${\mathrm{Hom}}_{\mathsf{Top}}(W\times X,Z)\cong {\mathrm{Hom}}_{\mathsf{Top}}(W,\mathrm{Map}(X,Z)).$$$\square$

Another nice property of the category $\mathsf{CG}$ is that product of quotient maps is still a quotient.

Proposition 5.15. Let $p_i:X_i\to Y_i,i=1,2$, be quotients in $\mathsf{CG}$. Then $p_1\times p_2:X_1\times X_2\to Y_1\times Y_2$ is a quotient.

Proof. We only need to show that if $p:X\to Y$ is a quotient map, then the induced map $q:X\times Z\to Y\times Z$ is a quotient. Here $X,Y,Z\in \mathsf{CG}$. Evidently, $q$ is surjective on sets. This is equivalent to show that for any map $f:Y\times Z\to W$, if $q\circ f$ is continuous, that $f$ is continuous. By the Exponential Law,$${\mathrm{Hom}}_{\mathsf{Top}}(X\times Z,W)={\mathrm{Hom}}_{\mathsf{Top}}(X,\mathrm{Map}(Z,W)).$$So $q\circ f$ is equivalent to a continuous map $X\to \mathrm{Map}(Z,W)$. Since $p:X\to Y$ is a quotient, this shows that $f$ corresponds to a continuous map $Y\to \mathrm{Map}(Z,W)$. Using the Exponential Law again,$${\mathrm{Hom}}_{\mathsf{Top}}(Y,\mathrm{Map}(Z,W))={\mathrm{Hom}}_{\mathsf{Top}}(Y\times Z,W).$$This implies the continuity of $f$.

$\square$

Compactly generated weak Hausdorff space

Definition 5.16. A space $X$ is weak Hausdorff if for every compact Hausdorff $K$ and every continuous map $f:K\to X$, the image $f(K)$ is closed in $X$.

Let $\mathsf{wH}$ denote the full subcategory of $\mathsf{Top}$ consisting of weak Hausdorff spaces. Let $\mathsf{CGWH}$ denote the full subcategory of $\mathsf{Top}$ consisting of compactly generated weak Hausdorff spaces.

Example 5.17. Hausdorff spaces are weak Hausdorff since compact subsets of Hausdorff spaces are closed. Therefore locally compact Hausdorff spaces are compactly generated weak Hausdorff spaces.

Proposition 5.18. The functor $k:\mathsf{wH}\to \mathsf{CGWH}$ is right adjoint to the embedding $i:\mathsf{CGWH}\subset \mathsf{wH}$. In other words, we have an adjoint pair

Proof. This follows from Proposition 5.3.

$\square$

Lemma 5.19. Let $X\in \mathsf{wH}$, $K$ compact Hausdorff and $f:K\to X$ continuous. Then $f(K)$ is compact Hausdorff.

Proof. $f(K)$ is compact and closed. Moreover, $f:K\to X$ is a closed map by hypothesis.

Let $x_1,x_2\in f(K)$ be two points. Since $X\in \mathsf{wH}$, $x_1,x_2$ are closed. Hence $f^{-1}(x_1),f^{-1}(x_2)$ are disjoint closed. Since $K$ is compact Hausdorff, there exists disjoint open subsets $U_1,U_2$ of $K$ such that $f^{-1}(x_i)\subset U_i$. Then $f(K)-f(K-U_i)$ give disjoint open neighborhoods of $x_i$.

$\square$

Remark 5.20. For $X$ weak Hausdorff, this lemma says that $Z\subset X$ is k-closed if and only if $Z\cap K$ is closed in $K$ for any compact Hausdorff subspace $K\subset X$.

Proposition 5.21. Let $X\in \mathsf{CG}$. Then $X$ is weak Hausdorff if and only if the diagonal subspace ${\Delta }_X=\{(x,x)|x\in X\}$ is closed in $X\times X$. Here $X\times X$ is the product in the category $\mathsf{CG}$.

Proof. Assume $X\in \mathsf{CGWH}$. We need to show that ${\Delta }_X$ is k-closed in $X\times X$. Let$$f=(f_1,f_2):K\to X\times X,f_i:K\to X,$$where $K$ is compact Hausdorff. Let$$L=f_1(K)\cap f_2(K),$$which is compact Hausdorff by Lemma 5.19. Consider the diagonal ${\Delta }_L$ in $L\times L$, which lies in the image$$L\to X\times X.$$Since $L$ is compact Hausdorff, ${\Delta }_L$ is a compact Hausdorff subspace of $X\times X$, hence closed in $X\times X$. It follows that $f^{-1}({\Delta }_X)=f^{-1}({\Delta }_L)$ is closed.

Conversely, assume $X\in \mathsf{CG}$ and ${\Delta }_X$ is closed in $X\times X$. Let $f:K\to X$ be a continuous map with $K$ compact Hausdorff. We need to show $f(K)$ is k-closed in $X$. Let $g:L\to X$ be any continuous map with $L$ compact Hausdorff. Consider$$(f,g):K\times L\to X\times X.$$Then$$g^{-1}(f(K))=(f,g{)}^{-1}({\Delta }_X),$$which is closed. This shows that $f(K)$ is k-closed in $X$, hence closed in $X$.

$\square$

Remark 5.22. Recall that $X\in \mathsf{Top}$ is Hausdorff if and only if ${\Delta }_X$ is closed in $X\times X$. This proposition says that $\mathsf{CGWH}$ relative to $\mathsf{CG}$ is the analogue of Hausdorff spaces relative to $\mathsf{Top}$.

Corollary 5.23. Let $\{X_i{\}}_{i\in I}$ be a family of objects in $\mathsf{CGWH}$. Then their product $\prod _{i\in I}{X}_i$ in $\mathsf{CG}$ also lies in $\mathsf{CGWH}$.

Proof. Let $X=\prod _{i\in I}{X}_i$ with ${\pi }_i:X\to X_i$. We need to show that the diagonal ${\Delta }_X$ is closed in $X\times X$. Let$${\pi }_i\times {\pi }_i:X\times X\to X_i\times X_i,D_i=({\pi }_i\times {\pi }_i{)}^{-1}({\Delta }_{X_i}).$$Since ${\Delta }_i$ is closed in $X_i\times X_i$, it follows that ${\Delta }_X={\bigcap }_{i\in I}{D}_i$ is closed in $X\times X$.

$\square$

Proposition 5.24. Let $X\in \mathsf{CG}$, and $E\subset X\times X$ be an equivalence relation on $X$. Then the quotient space $X/E$ by the equivalence relation $E$ lies in $\mathsf{CGWH}$ if and only if $E$ is closed in $X\times X$.

Proof. By Proposition 5.4, $X/E\in \mathsf{CG}$. We need to check the weak Hausdorff property.

Let $q:X\to Y=X/E$ denote the quotient map. By Proposition 5.15, the product$$q\times q:X\times X\to Y\times Y$$is also a quotient map. So ${\Delta }_Y$ is closed in $Y\times Y$ if and only if $(q\times q{)}^{-1}({\Delta }_Y)=E$ is closed in $X\times X$.

$\square$

Given $X\in \mathsf{CG}$, let $E_X$ denote the smallest closed equivalence relation on $X$. $E_X$ is constructed as the intersection of all closed equivalence relations on $X$. Then the quotient $X/E_X$ by the equivalence relation $E_X$ is an object in $\mathsf{CGWH}$. This construction is functorial, so defines a functor$$h:\mathsf{CG}\to \mathsf{CGWH}.$$

Proposition 5.25. The functor $h:\mathsf{CG}\to \mathsf{CGWH}$ is left adjoint to the inclusion $j:\mathsf{CGWH}\to \mathsf{CG}$. That is, we have an adjoint pairMoreover, $h$ preserves the subcategory $\mathsf{CGWH}$, i.e, $h\circ j$ is the identity functor.

Proof. Let $X\in \mathsf{CG},Y\in \mathsf{CGWH}$, and $f:X\to Y$ continuous. We need to show that $f$ factors through $X/E_X\to Y$. Consider$$f\times f:X\times X\to Y\times Y.$$Since ${\Delta }_Y$ is closed in $Y\times Y$, $(f\times f{)}^{-1}({\Delta }_Y)$ defines a closed equivalence relation on $X$. Therefore $E_X\subset (f\times f{)}^{-1}({\Delta }_Y)$. It follows that $f$ factors through $X\to X/E_X\to Y$.

$\square$

Theorem 5.26. The category $\mathsf{CGWH}$ is complete and cocomplete. Limits in $\mathsf{CGWH}$ inherit the limits in $\mathsf{CG}$. The colimits in $\mathsf{CGWH}$ are obtained by applying $h$ to the colimits in $\mathsf{CG}$.

Proof. Let $F\in \mathsf{Fun}(I,\mathsf{CGWH})$, then we need to show that$$\mathbf{colim}F=h\left(\mathbf{colim}(j\circ F)\right),j(\mathbf{lim}F)=\mathbf{lim}(j\circ F).$$

The statement about colimit follows from the fact that $h\circ j$ is the identity functor and $h$ perserves colimits. For the limit, let$$X=\prod _{i\in I}F(i),Y=\prod _{i\stackrel{f}{\to }j}F(j)$$be the products in $\mathsf{CG}$, which also lie in $\mathsf{CGWH}$ by Lemma 5.23. Consider two maps $g_1,g_2:X\to Y$, where$$g_1(\{x_i\})=\{x_j{\}}_{i\stackrel{f}{\to }j},g_2(\{x_i\})=\{f(x_i){\}}_{i\stackrel{f}{\to }j}.$$Then$$\mathbf{lim}(j\circ F)=\{x\in X|g_1(x)=g_2(x)\}=(g_1\times g_2{)}^{-1}({\Delta }_Y)$$is a closed subspace of $X$, hence also lies in $\mathsf{CGWH}$. It can be checked that this is the limit of $F$.

$\square$

Remark 5.27. The proof of the limit part of this theorem does not rely on $h$. It shows that $j:\mathsf{CGWH}\to \mathsf{CG}$ preserves all limits. The Adjoint Functor Theorem implies an abstract existence of $h$.

Proposition 5.28. Let $X,Y\in \mathsf{CGWH}$. Then $\mathrm{Map}(X,Y)\in \mathsf{CGWH}$.

Proof. We need to show that the diagonal ${\Delta }_{\mathrm{Map}(X,Y)}$ in $\mathrm{Map}(X,Y)\times \mathrm{Map}(X,Y)$ is closed. Let$$\begin{array}{rcl}e{v}_x:\mathrm{Map}(X,Y)& \to & Y,\\ f& \mapsto & f(x),\end{array}$$which is continuous. Then$${\Delta }_{\mathrm{Map}(X,Y)}=\bigcap _{x\in X}(e{v}_x\times e{v}_x{)}^{-1}({\Delta }_Y)$$is closed since ${\Delta }_Y$ is closed in $Y\times Y$.

$\square$

Combining Proposition 5.28, Proposition 5.11, Theorem 5.13, Proposition 5.14, we have the following theorem.

Theorem 5.29. Let $X,Y,Z\in \mathsf{CGWH}$. Then

1.	the evaluation map $\mathrm{Map}(X,Y)\times X\to Y$ is continuous;
2.	the composition map $\mathrm{Map}(X,Y)\times \mathrm{Map}(Y,Z)\to \mathrm{Map}(X,Z)$ is continuous;
3.	the Exponential Law holds, i.e., we have a homeomorphism$$\mathrm{Map}(X\times Y,Z)\cong \mathrm{Map}(X,\mathrm{Map}(Y,Z)).$$

Therefore $\mathsf{CGWH}$ is a full complete and cocomplete subcategory of $\mathsf{Top}$ that enjoys the Exponential Law.

We give a brief discussion on "subspace topology" in $\mathsf{CG}$ to end this section.

Let $X\in \mathsf{CG}$ and $A$ be a subset of $X$. The subspace topology on $A$ may not be compactly generated. We equip $A$ with a compactly generated topology by applying $k$ to the usual subspace topology. This will be called the subspace topology in the category $\mathsf{CG}$. When we write $A\subset X$, $A$ is understood as s subspace of $X$ with this compatly generated topology. It is clear that if $X\in \mathsf{CGWH}$, then $A\in \mathsf{CGWH}$. It can be checked that if $A$ is the intersection of an open and a closed subset of $X$, then the usual subspace topology on $A$ is already compactly generated, so these two notions of subspace coincide in this case.

This new notion of subspace satisfies the standard characteristic property in $\mathsf{CG}$: given $Y\in \mathsf{CG}$, a map $Y\to A$ is continuous if and only if it is continuous viewed as a map $Y\to X$.

Definition 5.30. In the category $\mathsf{CG}$, a map $i:A\to X$ in $\mathsf{CG}$ is called an inclusion if $A\to i(A)$ is a homeomorphism, where $i(A)$ is the image of $A$ with the compactly generated subspace topology from $X$.

Proposition 5.31. Let $X\stackrel{i}{\to }Y\stackrel{r}{\to }X$ be maps in $\mathsf{CGWH}$ such that $r\circ i=1_X$. Then $i$ is a closed inclusion and $r$ is a quotient map.

Proof. It is clear that $i$ is an inclusion and $r$ is a quotient. We show $i(X)$ is a closed inclusion. Consider$$(i\circ r,1_Y):Y\to Y\times Y.$$Let ${\Delta }_Y\subset Y\times Y$ be the diagonal which is closed. Then $i(X)=(i\circ r,1_Y{)}^{-1}({\Delta }_Y)$ is also closed.

$\square$

Proposition 5.32. Let $X,Y,Z\in \mathsf{CGWH}$ and $i:X\to Y$ is an inclusion. Then $i\times 1_Z:X\times Z\to Y\times Z$ is also an inclusion. If $i$ is closed, then so is $i\times 1_Z$.

We will often need the notion of a pair. Given $X,Y\in \mathsf{CGWH}$, and subspaces $A\subset X,B\subset Y$, we let$$\mathrm{Map}((X,A),(Y,B))=\{f\in \mathrm{Map}(X,Y)|f(A)\subset B\}$$be the subspace of $\mathrm{Map}(X,Y)$ that maps $A$ to $B$. It fits into the following pull-back diagram

In our later discussion on homotopy theory, we will mainly work with $\mathsf{CGWH}$. In particular, a space there always means an object in $\mathsf{CGWH}$. All the limits and colimits are in $\mathsf{CGWH}$. For example, given $X,Y\in \mathsf{CGWH}$, their product $X\times Y$ always means the categorical product in $\mathsf{CGWH}$. Subspace refers to the compacted generated subspace topology.

To simplify notations, we will write$$\underline{\mathcal{T}}=\mathsf{CGWH}\text{and}\underline{h\mathcal{T}}$$for the category $\mathsf{CGWH}$ and the quotient category of $\underline{\mathcal{T}}$ by homotopy classes of maps respectively.

We will also need the category of pointed spaces.

Definition 5.33. We define the category $\underline{{\mathcal{T}}_{\star }}$ of pointed spaces, where

•	an object $(X,x_0)$ is a space $X\in \underline{\mathcal{T}}$ with a based point $x_0\in X$ and
•	morphisms are based continuous maps that map based point to based point$${\mathrm{Hom}}_{\underline{{\mathcal{T}}_{\star }}}((X,x_0),(Y,y_0))=\mathrm{Map}((X,x_0),(Y,y_0)).$$

We will write$${\mathrm{Map}}_{\star }(X,Y)=\mathrm{Map}((X,x_0),(Y,y_0))$$when base points are not explicitly mentioned. ${\mathrm{Map}}_{\star }(X,Y)$ is viewed as an object in $\underline{{\mathcal{T}}_{\star }}$, whose base point is the constant map from $X$ to the base point of $Y$.

The following theorem follows from the analogue for $\underline{\mathcal{T}}$ described above.

Theorem 5.34. The category $\underline{{\mathcal{T}}_{\star }}$ is complete and cocomplete. Let $X,Y,Z\in \underline{{\mathcal{T}}_{\star }}$. Then

1.	the evaluation map ${\mathrm{Map}}_{\star }(X,Y)\wedge X\to Y$ is continuous;
2.	the composition map ${\mathrm{Map}}_{\star }(X,Y)\wedge {\mathrm{Map}}_{\star }(Y,Z)\to {\mathrm{Map}}_{\star }(X,Z)$ is continuous;
3.	the Exponential Law holds, i.e., we have a homeomorphism$${\mathrm{Map}}_{\star }(X\wedge Y,Z)\cong {\mathrm{Map}}_{\star }(X,{\mathrm{Map}}_{\star }(Y,Z)).$$

Here $\wedge$ is the smash product$$X\wedge Y=\frac{X\times Y}{X\times \{y_0\}\cup \{x_0\}\times Y}.$$