Poincare polynomial of a space and of a pair over a field

Let $F$ be a field (Field) and let $(X,A)$ be a pair of spaces whose relative singular homology $H_k(X,A;F)$ (The singular chain complex and singular homology, Relative singular homology) is finite-dimensional over $F$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis) for every $k$ and vanishes for all sufficiently large $k$. Write $b_k(X,A;F):={\dim }_F{H}_k(X,A;F)$ and define the relative Poincare polynomial over $F$ by $P_{X,A}(t):={\sum }_k{b}_k(X,A;F)t^k\in \mathbb{Z}[t]\mathrm{.}$ When $A=\mathrm{\varnothing }$, so that $H_k(X,\mathrm{\varnothing };F)=H_k(X;F)$, the polynomial $P_X(t):=P_{X,\mathrm{\varnothing }}(t)={\sum }_k{b}_k(X;F)t^k$ is the Poincare polynomial of $X$ over $F$.

Under ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), for a closed smooth $n$-manifold the hypotheses hold: all $b_k$ are finite and vanish for $k>n$. The coefficients are the $F$-Betti numbers and generally depend on $F$ when $H_{\ast }(X;\mathbb{Z})$ has torsion.

• Well-definedness. Each coefficient $b_k(X,A;F)$ is the dimension of a finite-dimensional $F$-vector space, an invariant of that space independent of bases (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis); the sum is finite by the vanishing hypothesis.
• The finiteness claim for closed manifolds. Under ${\mathrm{A}\mathrm{C}}_{\omega }$, an excellent Morse function exists by Adapted excellent Morse functions exist on compact cobordisms applied to the empty-face triad. Finiteness is then proved on this page by the handle chain complex of a Morse function: the sublevel filtration supplies a finite-dimensional homology computation, and an index-ordered handle presentation supplies a finite CW model homotopy equivalent to $M$, so $H_k(M;F)$ is finite-dimensional for every $k$ and vanishes for $k>\dim M$. The definition itself is conditional on the stated hypotheses, so no circularity arises.
• The empty case. If $X=\mathrm{\varnothing }$ then $H_k(\mathrm{\varnothing };F)=0$ for all $k$ and $P_{\mathrm{\varnothing }}(t)=0$; if $A=X$ then all relative homology vanishes and $P_{X,X}(t)=0$. Both are consistent with the sum over an empty family of nonzero terms.