Proving the Weak Continuity Property is Equivalent to the Continuity Property (in Cohomology) In Edwin H. Spanier's Algebraic Topology, he describes both a "continuity property" and a "weak continuity property" for cohomology theories. (He actually defines these in terms of pairs $(X,A)$ where $A$ is some subspace of $X$, but I don't care about this; I just want to understand this for spaces.)
First of all, if $\lbrace X_i,p_{ij} \rbrace_{i,j\in I}$ is an inverse system of compact Hausdorff spaces and $X:=\varprojlim X_i$, then for any cohomology theory $H^*(\bullet)$ the projection maps $p_{ij}:X_j\to X_i$ induce maps $p_{ij}^*:H^k(X_i)\to H^k(X_j)$ which form a direct system of groups/modules; furthermore the maps $p_i:X\to X_i$ induce maps $p_i^*:H^k(X_i)\to H^k(X)$ which are compatible with that direct system, so we obtain a map $p^*:\varinjlim H^k(X_i)\to H^k(X)$. If the map $p^*$ is an isomorphism, then the cohomology theory is called "continuous".
Next, if every $X_i$ is a subspace of some ambient space (let's say $Y$) and every projection map $p_{ij}:X_j \to X_i$ is just the inclusion of $X_j$ as a subspace of $X_i$, then actually $X=\underset{i\in I}{\bigcap}X_i$ and each $p_i:X\to X_i$ is also inclusion. I'll call this situation a "nested system", as I think I've seen that terminology for it elsewhere. If $p^*$ is an isomorphism just for nested systems, then the cohomology theory is called "weakly continuous".
Spanier says that "it is not hard to prove that the continuity property is equivalent to the weak continuity property". Well, one direction (continuity implies weak continuity) is trivial, but I have a gap in attempting to prove the other direction.
Let's say a cohomology theory satisfies weak continuity, and I have some inverse system of compact Hausdorff spaces. What appears to be the obvious way to reconstruct this as a nested system is to make the ambient space $Y:=\underset{i\in I}{\prod } X_i$ since $X$ is already constructed as a subspace of this, and then make a nested system $\lbrace Y_i \rbrace_{i \in I}$ by fixing some basepoint $b\in X$ and making each $Y_i$ equal to the product of $\underset{i\leq j}{\prod}X_{j}$ with the "$X_k$" components of the base point for all $k<i$. With this, I can make the natural projection maps $\pi_j:\underset{i\in I}{\prod } X_i\to X_j$, which are surjective, give me an injective map $\pi^*:\varinjlim H^*(X_i)\to\varinjlim(Y_i)\cong H^*(X)$ (where the last isomorphism is because $H^*(\bullet)$ is weakly continuous). But here I get stuck, because I can't see any way to either show that this map is also surjective or make an injective map going the other way. Any map I can easily think of seems to be going the wrong direction to complete the needed isomorphism. So, what am I missing?
 A: In excercise 6.C.1 Spanier gives the following hint (he does it for pairs, here we only consider the absolute case):

Prove that $X$ can be embedded in a space in which it is a directed intersection of compact Hausdorff spaces $X'_i$ where $X'_i$ has the same homotopy type as $X_i$. Hint: For each $k \in I$ embed $X_k$ in a contractible compact Hausdorff space $Y_k$, for example a cube, and let $X'_i \subset \prod_{k \in I} Y_k$ be defined as the set of all points $(y_k)$ with $y_i \in X_i$ and $y_j = p_{ji}(y_i)$ for $j \le i$. All other coordinates $y_j$ are arbitrary.

Let us analyze this. W.l.o.g. we may assume $X_k \subset Y_k$ as a genuine subset with inclusion map $\iota_k : X_k \to Y_k$. Let $P_i = \prod_{k \le i} Y_k$ and let $\pi_k : P_i \to Y_k$ denote the projections for $k \le i$. Moreover let $Q_i = \prod_{k \not\le i} Y_k$, and let $q_k :  P = \prod_{j \in I} Y_j \to Y_k$ denote the projections.

*

*There is a unique map $\phi_i : X_i \to P_i$ such that $\pi_k \phi_i = \iota_k p_{ki}$. It is injective since $\pi_i \phi_i = \iota_i$. Because $X_i$ is compact  and $P_i$ is Hausdorff, $\phi_i$ is an embedding and thus gives a homeomorphism $\phi'_i : X_i \to X''_i = \phi_i(X_i)$. The inverse homeomorphism is nothing else than the restriction $r_i : X''_i \to X_i$ of $\pi_i$. Clearly $X'_i = X''_i \times Q_i$. Since $Q_i$ is contractible, the map $R_i = r_i q_i : X'_i \to X_i$ with projection $q_i : X''_i \times Q_i \to X''_i$ is a homotopy equivalence. By construction, $R_i$ is nothing else than the restriction of $q_i$ to $X'_i$.


*For $l \le i$ we have $X'_i \subset X'_l$. This is clear because $X_i$ is the set of all  $(y_k)$ with $y_i \in X_i$ and $y_j = p_{ji}(y_i)$ for $j \le i$. Each such $(y_j)$ has $y_l = p_{li}(y_i) \in X_l$ and $p_{jl}(y_l) = p_{jl}(p_{li}(y_i)) = p_{ji}(y_i) =y_j$ for $j \le l$, thus $(y_j) \in X'_l$. By $e_{li} : X'_i \to X'_l$ we denote the inclusion map.


*We have
$$\varprojlim X'_i = \bigcap_{i \in I} X'_i = \{(y_j) \mid y_j \in X_j \text{ for all } j, p_{kj}(y_j) = y_k \text{ for all } k \le j \}  \\=  \{(y_j) \in \prod_{j \in I} X_j \mid p_{kj}(y_j) = y_k \text{ for all } k \le j \} = \varprojlim X_i = X .$$


*For $l \le i$ we have $R_l e_{li} = p_{li} R_i$. In fact, for $(y_j) \in X'_i$ we have
$$R_l e_{li}((y_j)) = R_l((y_j)) = q_l((y_j)) = y_l = p_{li}(y_i) = p_{li}(q_i((y_j))) =  p_{li}(R_i((y_j))) .$$
Thus the $R_i$ form a map of inverse systems from $(X'_i,e_{ji})$ to $(X_i,p_{ji})$ such that all $R_i^*$ are isomorphisms of cohomology groups. The desired result follows.
