Skeletal filtration for generalized cohomology

Let $X$ be a finite CW complex with skeleta $X^p$, and let $h$ be the CW-pair cohomology theory associated with a reduced generalized cohomology theory $\tilde{h}$ by Reduced and unreduced generalized cohomology theories correspond. Use the conventions $X^p=\mathrm{\varnothing }(p<0),X^p=X(p\ge \dim X),$ so that $X^{-1}=\mathrm{\varnothing }$. The skeletal filtration of $h^n(X)$ is the decreasing filtration $F^p{h}^n(X):=\ker (h^n(X)\to h^n(X^{p-1})),p\in \mathbb{Z}\mathrm{.}$

By the long exact sequence (LES) of the pair $(X,X^{p-1})$ and the fact that a relative group maps onto the kernel of the restriction, the same subgroup is $F^p{h}^n(X)=\mathrm{im}(h^n(X,X^{p-1})\to h^n(X)),$ the image being the kernel of the next restriction by exactness. The filtration is decreasing, $F^{p+1}{h}^n(X)\subseteq F^p{h}^n(X)$, because the restriction to $X^{p-1}$ factors through the restriction to $X^p$. For a finite CW complex it is bounded and exhaustive: $F^p{h}^n(X)=h^n(X)$ for $p\le 0$, since $h^n(\mathrm{\varnothing })=0$, and $F^p{h}^n(X)=0$ for $p>\dim X$, since then $X^{p-1}=X$ and the restriction is the identity. No completeness or infinite dimension is asserted.

Compare Loizides, §3, printed pp. 4–7, where for a finite CW complex the filtration is written $F^m{h}^n(X)=\ker (h^n(X)\to h^n(X^m))$, the indexing here being shifted by one so that $F^p{h}^n(X)$ corresponds to the classes vanishing on the $(p-1)$-skeleton; see also Theorem 3.4 and Remark 3.5 there.