Postnikov towers exist for connected CW complexes

Fix $n\ge 1$ and put $Z_n=X$. Suppose $Z_{t-1}$ has been constructed for $t>n$. Choose a based sphere map $S^t\to Z_{t-1}$ representing every element of ${\pi }_t(Z_{t-1})$, and attach one $(t+1)$-cell along each map to form $Z_t$. The relative pair has only $(t+1)$-cells, so [F1] preserves every ${\pi }_i$ with $i<t$ and makes ${\pi }_t(Z_{t-1})\to {\pi }_t(Z_t)$ surjective.

In the relative homotopy exact sequence, each selected attaching map is the boundary of its characteristic-disk class by [F2]. Because all elements were selected, the boundary ${\pi }_{t+1}(Z_t,Z_{t-1})\to {\pi }_t(Z_{t-1})$ is surjective. Exactness and the vanishing of ${\pi }_t(Z_t,Z_{t-1})$ from [F1] therefore give ${\pi }_t(Z_t)=0$.

Define $P_{n}X={\bigcup }_{t>n}{Z}_t$ and let $p_n$ be the inclusion of $X$. Every added cell has dimension $t+1\ge n+2$. For $i\le n$, all inclusions preserve ${\pi }_i$ by [F1]. For fixed $i>n$, Step 2.1 kills ${\pi }_i$ at stage $Z_i$, and later cells have dimension at least $i+2$, so [F1] prevents its reappearance.

A sphere representative in the union has image in a finite subcomplex by [F3], hence in one $Z_t$; the same holds for a disk nullhomotopy. It follows in both the surjective and injective directions that ${\pi }_i(P_{n}X)$ is the sequential colimit of the stage groups. Step 3.1 thus gives

Carry out Steps 1.1--4.1 for every $n\ge 1$ under [A1], and put $P_{0}X=\ast$. Suppose $p_{n-1}:X\to P_{n-1}X$ is fixed. The target has no homotopy above degree $n-1$, while every relative cell of $(P_{n}X,X)$ has dimension at least $n+2$. Extending $p_{n-1}$ one cell at a time encounters an attaching sphere of dimension at least $n+1$, whose class in the target is zero. Simultaneous fillers give $q_n:P_{n}X\to P_{n-1}X$ with $q_n{p}_n=p_{n-1}$.

If $q_n^{\mathrm{\prime }}$ is another such extension, regard a homotopy rel $X$ as an extension over the relative prism cells. Their dimensions are one greater than those of $(P_{n}X,X)$, so every obstruction again lies above degree $n-1$ and vanishes. Thus $q_n\simeq q_n^{\mathrm{\prime }}$ rel $X$. Together with the unique map to $P_{0}X$, these maps form the claimed tower.

Empty higher homotopy groups merely yield empty attachment families, and the trivial group requires no representative. Connectedness supplies a common component and based groups throughout. All arbitrary family selections occur in Steps 1.1 and 5.1--6.1 and are covered by [A1]; the limit argument itself is the individual finite-support argument of [F3]. The construction gives no comparison from $X$ to an ordinary or homotopy inverse limit, so no convergence has been smuggled in. $\mathrm{\square }$