Cw Complexes and Cellular Homology

The defining equivalence relation identifies only boundary points of $D^n$ with points of $X$ (and may identify boundary points with one another); no point of ${\mathring{D}}^n$ is equivalent to a distinct point.

The saturated open set ${\mathring{D}}^n$ has quotient homeomorphic to its image, and its complement in the closed-cell image is the image of $S^{n-1}$. Thus the image is open relative to the attached space, while step 1.1 deliberately says nothing about the boundary.

Let ${\chi }_e:D^m\to X$ be the characteristic map of an $m$-cell. If $m\le n$, then ${\chi }_e^{-1}(X^n)=D^m$. If $m>n$, this inverse image lies in $S^{m-1}$ and equals the inverse image there of the already closed lower skeleton $X^n\subseteq X^{m-1}$; induction on $m$ makes it closed in $S^{m-1}$ and hence in $D^m$. The weak-topology clause now makes $X^n$ closed in $X$.

The open-cell interiors are disjoint by the cellular partition, and The interior of an attached cell embeds openly in its closure identifies precisely the new interior points after $X^{n-1}$. This gives the asserted difference.

If $K$ met infinitely many cells, use [A1] recursively on the nonempty set of cells remaining after finitely many earlier choices, obtaining distinct met cells and points $x_i\in K$ in them. Put $S=\{x_i:i\ge 1\}$. Induction over the skeleta shows that $S\cap X^n$ is closed in $X^n$: on the boundary of each characteristic $n$-disk this follows from the induction hypothesis, and the disk interior contains at most one $x_i$. The weak topology then closes the induction.

The same argument applies to every subset of $S$, so $S$ is a closed discrete subspace of $K$. An infinite discrete space is not compact, contradicting compactness of the closed subspace $S\subseteq K$.

The characteristic maps are part of the supplied CW structure. Thus the given compact space and continuous map satisfy the hypotheses of [L1].

Apply [L1] to obtain a finite CW subcomplex containing $f(K)$, including the empty-image case. The supplier uses canonical coordinate minima to prove finite cell support without a choice premise.

First establish the needed quotient-cylinder mapping criterion. Let $q:E\to B$ be any quotient map and let $h:B\times I\to Y$ be a function such that $h\circ (q\times {\mathrm{i}\mathrm{d}}_I)$ is continuous. For each $b$, surjectivity supplies a representative $e$ and shows that the path $t\mapsto h(b,t)$ is continuous. No simultaneous selection of representatives is needed. The continuous transpose upstairs is $\widehat{h}\circ q:E\to P$. For every open $V\subseteq P$, its inverse image is $q^{-1}[{\widehat{h}}^{-1}[V]]$, so the quotient criterion makes ${\widehat{h}}^{-1}[V]$ open. Thus $\widehat{h}$ and then $h$ are continuous. No closedness hypothesis on $q$ is used.

For $n\ge 1$, put $T_n=(D^n\times \{0\})\cup (S^{n-1}\times I)$. For $u\in D^n$ and $t\in I$, define $a=\max \{1-t\mathrm{/}2,\mathrm{\parallel }u\mathrm{\parallel }\}$, $\lambda =1\mathrm{/}a$, and $r_n(u,t)=(\lambda u,2+\lambda (t-2))$. Since $1\mathrm{/}2\le a\le 1$, this formula is continuous and defined on the entire cylinder. Moreover $1\le \lambda \le 2\mathrm{/}(2-t)$, so its height lies in $[0,t]$, and $\lambda \mathrm{\parallel }u\mathrm{\parallel }\le 1$. If $a=1-t\mathrm{/}2$ the height is zero, while if $a=\mathrm{\parallel }u\mathrm{\parallel }$ the norm of the first coordinate is one; hence the image lies in $T_n$. If $t=0$ or $\mathrm{\parallel }u\mathrm{\parallel }=1$, then $a=1$ and the point is fixed. Thus $r_n$ is a retraction onto $T_n$. For $n=0$ set $r_0(\ast ,t)=(\ast ,0)$.

We specify compatible continuous maps $H_n:X^n\times I\to Y$, with $H_n(x,0)=g(x)$ and $H_n=H$ on $(A\cap X^n)\times I$. On the discrete zero skeleton prescribe $H_0(v,t)=H(v,t)$ for $v\in A$ and $H_0(v,t)=g(v)$ otherwise. Every vertex cylinder is open, so this is continuous.

We justify the infinite-stage continuity using the weak topology on $X$, not an assumed weak topology on $X\times I$. Each characteristic map $\varphi :D^n\to \bar{e}$ is a quotient map. Indeed it is a continuous surjection; if $F\subseteq D^n$ is closed, it is compact. Its image is compact by pulling any open cover of $\varphi [F]$ back to $F$ and retaining the images' finitely many covering members. This open-cover argument works for arbitrary targets, without a metrizability hypothesis on $\bar{e}$. Since $\bar{e}$ is a subspace of the Hausdorff space $X$, $\varphi [F]$ is closed there. Thus $\varphi$ is closed; if ${\varphi }^{-1}[C]$ is closed, then $C=\varphi [{\varphi }^{-1}[C]]$ is closed, proving the quotient criterion.

Given such $H_{n-1}$ for $n\ge 1$, let ${\varphi }_{\alpha }:D^n\to X^n$ be a supplied characteristic map and $f_{\alpha }:S^{n-1}\to X^{n-1}$ its boundary map. If its open cell belongs to $A$, use $K_{\alpha }(u,t)=H({\varphi }_{\alpha }(u),t)$; this is defined on the whole disk cylinder because the closed cell lies in $A$. For every other cell define $b_{\alpha }:T_n\to Y$ by $b_{\alpha }(u,0)=g({\varphi }_{\alpha }(u))$ and $b_{\alpha }(u,t)=H_{n-1}(f_{\alpha }(u),t)$ for $u\in S^{n-1}$. These two continuous prescriptions agree on the corner and their domains form a two-piece closed cover of $T_n$, so $b_{\alpha }$ is continuous. Set $K_{\alpha }=b_{\alpha }\circ r_n$.

The maps $H_{n-1}$ and all $K_{\alpha }$ agree on the attaching identifications, including for cells belonging to $A$, and so determine a function $H_n:X^n\times I\to Y$. The map from $(X^{n-1}\bigsqcup {\coprod }_{\alpha }{D}^n)\times I$ is continuous: each summand cylinder is open and carries the prescribed continuous map. The attachment map $X^{n-1}\bigsqcup {\coprod }_{\alpha }{D}^n\to X^n$ is quotient, so step 1.1 proves continuity of $H_n$. Its time-zero values are $g$, it extends $H_{n-1}$, and it equals $H$ on $(A\cap X^n)\times I$, since every cell of this intersection is either in the previous skeleton or among the cells on which $K_{\alpha }$ was prescribed from $H$.

These formulas define the successor map uniquely from the preceding stage; they do not choose one extension from a nonempty set of extensions. Apply ordinary recursion, recording the stage number together with the partial map, to obtain all $H_n$. The state space can be taken to be the set of valid stage-number/partial-map pairs, with the successor just constructed. Define $\tilde{H}(x,t)=H_n(x,t)$ whenever $x\in X^n$. Compatibility makes this independent of $n$, gives $\tilde{H}(x,0)=g(x)$, and gives $\tilde{H}{\mathrm{\mid }}_{A\times I}=H$.

Every path $t\mapsto \tilde{H}(x,t)$ is continuous by some $H_n$, so the transpose $G:X\to P$ is well defined. For each characteristic map $\varphi :D^n\to X$, the map $\tilde{H}\circ (\varphi \times {\mathrm{i}\mathrm{d}}_I)$ is continuous by construction, and [L3] makes $G\circ \varphi$ continuous. By the quotient property from step 1.4, $G{\mathrm{\mid }}_{\bar{e}}$ is continuous. Consequently, for every closed $C\subseteq P$, the set $G^{-1}[C]$ meets every closed cell in a relatively closed set. The CW weak topology implies $G^{-1}[C]$ is closed in $X$. Thus $G$ is continuous. This argument permits arbitrary families of cells and unbounded dimension.

The exponential law applied once more, now to $G:X\to P$, proves that $\tilde{H}:X\times I\to Y$ is continuous. Its restrictions established in step 4.1 solve every homotopy-extension problem in the statement, which is the asserted cofibration property.

Necessity follows by restriction. Conversely, $f{\mathrm{\mid }}_{\bar{e}}$ is continuous because $\bar{e}$ lies in the skeleton of its dimension.

For closed $C\subseteq Y$, each $f^{-1}(C)\cap \bar{e}$ is closed in $\bar{e}$ by step 1.1; the weak-topology clause in CW complex with closure finiteness and weak topology makes $f^{-1}(C)$ closed.

The image $L$ is compact in the open-cover sense. Indeed, the inverse images of any open cover of $L$ cover $K$; a finite subcover of $K$ yields a finite subcover of $L$ by the same covering members. This argument does not impose a metric on the target. Since $X$ is Hausdorff, [F2] makes $L$ closed in $X$.

Every nonempty compact subset $T\subseteq {\mathbb{R}}^d$ has a uniquely specified lexicographically least point. For $d\ge 1$, minimize its first coordinate using [F5] and restrict to the minimum level set. That set is nonempty, closed in $T$ and compact by [F3]. Minimize the next coordinate on it and continue through the finite ordered coordinate set. After $d$ steps all coordinates are fixed, and the nonempty final set is a singleton. Each minimum value and each level set is unique; no minimizing point is selected until the final singleton. For $d=0$, the sole possible nonempty subset of ${\mathbb{R}}^0$ is already a singleton. This is a finite prescription defined for every such $T$, not a family of arbitrary existential choices.

Let $e$ be a positive-dimensional open cell meeting $L$, with characteristic map ${\chi }_e:D^d\to X$. For $r\ge 0$ let $B_{d,r}$ be the concentric closed ball of radius $1-1\mathrm{/}(r+2)$ in $D^d$. These balls exhaust its interior. Thus some $B_{d,r}$ meets ${\chi }_e^{-1}(L)$; let $r_e$ be the least such integer. The set $T_e=B_{d,r_e}\cap {\chi }_e^{-1}(L)$ is a nonempty compact subset of ${\mathbb{R}}^d$: it is closed in the compact ball by step 1.1, continuity and [F3], [F4]. Let $v_e$ be its uniquely specified point from step 1.2 and set $x_e={\chi }_e(v_e)\in L\cap e$. For an occupied zero-cell use that point itself. The least integer, the finite sequence of coordinate minima, and the supplied characteristic map specify $x_e$ uniquely for every occupied cell; the resulting function is defined by this formula on the set of occupied cells.

Put $S=\{x_e:e\cap L\mathrm{\ne }\mathrm{\varnothing }\}$. Distinct occupied cells give distinct points, since their interiors are disjoint. For every subset $T\subseteq S$ and every closed cell $\bar{a}$, closure finiteness in [F1] says that $\bar{a}$ meets only finitely many open cells. Hence $T\cap \bar{a}$ is finite, with at most one point from each of those cells. A finite subset of a Hausdorff space is closed: singleton complements are open by the Hausdorff separation axiom, and finite unions of closed sets are closed. The weak topology in [F1] now makes $T$ closed in $X$. In particular $S$ is closed in $X$, hence closed in the compact $L$. By [F3], $S$ is compact.

For $s\in S$, the set $S\setminus \{s\}$ is closed in $X$ by step 3.1. Its complement intersects $S$ in $\{s\}$, so $S$ is discrete. Its singleton cover is an open cover of $S$ and therefore has a finite subcover. Thus $S$ is finite, without first extracting a countably infinite subset from an arbitrary infinite set. The bijection $e\mapsto x_e$ from occupied cells to $S$ shows that only finitely many cells meet $L$.

If there are occupied cells, start with their finite set. Add every cell meeting the boundary of a cell already in the set, and repeat downward in dimension. At each stage only finitely many cells are added by closure finiteness [F1]. A cell boundary lies in the preceding skeleton, so the dimensions strictly decrease along every newly required boundary chain. The finite starting set has a maximum dimension $N$, and after at most $N$ such downward stages no more are required. The union of these cells contains the entire closure of each member, hence is a finite CW subcomplex by [F1]. It contains $L$, because every point of $L$ belongs to an occupied open cell.

If $K$ or $L$ is empty, the empty subcomplex suffices and no minima are taken. For a point image, the closure process starts at its one occupied cell; it need not itself be a zero-cell. Zero-dimensional cells and the zero-dimensional Euclidean coordinate space were handled without a norm or empty-coordinate minimum. The radii in step 2.1 are strictly between zero and one and approach one, so all chosen preimages lie in cell interiors and no boundary point is mistaken for a point of that open cell. Nonregular characteristic maps cause no problem, since only their interior restrictions are used for the selected points. Steps 1.2 and 2.1 specify every selection uniquely, while steps 4.1 and 5.1 use compactness and finite closure operations; no AC is used anywhere.

If there are no $n$-cells, $X^n=X^{n-1}$ and both sides are zero. For $n=0$, $X^0$ is discrete and $X^{-1}=\mathrm{\varnothing }$. Every singular simplex has constant image, so its singular chain complex splits over the vertices. On one point its degree-$k$ chain group is $G$ and its boundary is multiplication by ${\sum }_{i=0}^k(-1{)}^i$, which is zero for odd $k$ and the identity for positive even $k$; hence its homology is $G$ in degree zero and zero otherwise. This gives the asserted direct sum without an additivity axiom. Hence assume $n\ge 1$ and that an $n$-cell exists. Its attaching sphere makes $X^{n-1}$ nonempty.

We verify the good-pair hypothesis for $(X^n,X^{n-1})$. For each $n$-cell use its supplied characteristic map ${\chi }_e:D^n\to X^n$, and let $V=X^{n-1}\cup {\bigcup }_e{\chi }_e\{u\in D^n:\mathrm{\parallel }u\mathrm{\parallel }>1\mathrm{/}2\}\mathrm{.}$ The preimage of $V$ in each characteristic disk of dimension below $n$ is the whole disk, and in each $n$-disk it is the relatively open outer annulus $\{\mathrm{\parallel }u\mathrm{\parallel }>1\mathrm{/}2\}$; the CW weak topology therefore makes $V$ open. On each such annulus set $H_t({\chi }_e(u))={\chi }_e((1-t)u+tu\mathrm{/}\mathrm{\parallel }u\mathrm{\parallel }),$ and fix $X^{n-1}$ pointwise. The formula is continuous on each characteristic-disk cylinder and fixes the boundary sphere where $\mathrm{\parallel }u\mathrm{\parallel }=1$, so it agrees under every attaching identification, even for nonregular cells. Here is the product-continuity check for arbitrary cell sets. For each point of $V$, its track is continuous; transpose these tracks to a function $\widehat{H}:V\to C(I,V)$ with the compact-open topology. By The exponential law: for a locally compact metric $X$ and any spaces $Z$ and $Y$, transposition is a bijection between $C(X\times Z,Y)$ and $C(Z,C(X,Y))$ with the compact-open topology, the restriction of $\widehat{H}$ to every characteristic disk's inverse image of $V$ is continuous. The restricted characteristic maps jointly give the weak topology on the open subspace $V$: a subset of $V$ is open exactly when its inverse image in each disk is open. Hence $\widehat{H}$ is continuous, and the same interval exponential law untransposes it to a continuous $H:V\times I\to V$. At $t=1$ it lands in $X^{n-1}$; thus $V$ deformation retracts onto that closed subcomplex. By [F3], $H_k(X^n,X^{n-1};G)\cong {\tilde{H}}_k(W;G),W=X^n\mathrm{/}{X}^{n-1}={\bigvee }_e(D_e^n\mathrm{/}{S}_e^{n-1})\mathrm{.}$ The quotient is a wedge of $n$-spheres with common basepoint $b$ by the attaching presentation, and the isomorphism is induced by the quotient map.

First let $W_J$ be a finite wedge and split off one sphere $S_j^n$, leaving the subwedge $U$ that contains the common basepoint. This is again a skeletal attachment pair: the open neighborhood $U\cup {\chi }_j\{\mathrm{\parallel }u\mathrm{\parallel }>1\mathrm{/}2\}$ and the radial homotopy of step 1.2 retract onto $U$. Therefore [F3] applies, and the quotient $W_J\mathrm{/}U\cong S_j^n$ gives $H_k(W_J,U;G)\cong {\tilde{H}}_k(S_j^n;G)$. The inclusion $U\hookrightarrow W_J$ has a retraction collapsing $S_j^n$; hence its homology map is injective. In the pair long exact sequence this also forces the connecting map into $H_{k-1}(U;G)$ to be zero. Thus, for $k>0$, the sequence from $H_k(U;G)$ to $H_k(W_J;G)$ to ${\tilde{H}}_k(S_j^n;G)$ is short exact. It splits by the inclusion of $S_j^n$, since projection to the quotient is the identity on that sphere. Induction over the finite set $J$, together with sphere homology, gives $H_k(W_J;G)={⨁}_{j\in J}G$ for $k=n$ and zero for other positive $k$. Every finite wedge is path connected, including the empty wedge interpreted as a point, so ${\tilde{H}}_0(W_J;G)=0$.

The choice-free compact-image lemma puts the image of each singular simplex in the arbitrary wedge $W$ inside a finite CW subcomplex, hence inside a finite subwedge. Every singular chain is a finite sum of simplices, so it too is supported in a finite subwedge. A cycle in $W$ supported there is a cycle there, since singular chain inclusion is injective. If a finite supported cycle bounds in $W$, its bounding chain also lies in some finite subwedge; enlarging both finite supports makes it a boundary in one finite subwedge. Therefore the finite-wedge calculations of step 2.1 pass elementwise to $W$: classes have finite support, and any relation is already a finite-wedge relation. This gives ${\tilde{H}}_n(W;G)\cong {⨁}_{e_{\alpha }^n}G$ and zero in every other degree, including reduced degree zero because $n\ge 1$ and $W$ is path connected. Composing with the quotient-induced isomorphism of step 1.2 proves the stated natural isomorphism, without a choice premise.

For $n=1$ the composite is zero because $d_0=0$ by definition.

Let $n\ge 2$. Write $d_n=j_{n-1}{\mathrm{\partial }}_n,d_{n-1}=j_{n-2}{\mathrm{\partial }}_{n-1},$ where $j_{n-1}:H_{n-1}(X^{n-1})\to H_{n-1}(X^{n-1},X^{n-2})$ is the quotient map. In the long exact sequence of the pair $(X^{n-1},X^{n-2})$, $j_{n-1}$ is immediately followed by ${\mathrm{\partial }}_{n-1}$.

Exactness gives ${\mathrm{\partial }}_{n-1}{j}_{n-1}=0$, and hence $d_{n-1}{d}_n=j_{n-2}({\mathrm{\partial }}_{n-1}{j}_{n-1}){\mathrm{\partial }}_n=0.$ Together with step 1.1 this covers every $n\ge 1$.

Its relative homology vanishes outside degree $n+1$ by Relative homology of consecutive CW skeleta.

In the pair long exact sequence, the two relative groups adjacent to $H_k(X^n)$ vanish for $k<n$, and only the outgoing one need vanish for $k=n$; exactness gives respectively an isomorphism and a surjection.

A singular chain has finitely many simplices, so its image is compact; The image of a compact space lies in a finite CW subcomplex places it in one finite subcomplex and hence in some skeleton. Thus every homology class is in the image.

If a class from $H_k(X^n)$ dies in $H_k(X)$, a finite singular bounding chain has compact image and lies in some $X^m$. It already kills the class there, proving injectivity of the colimit map.

For every finite $m\ge 0$, $H_k(X^m)=0$ when $k>m$: begin with the discrete $X^0$ case of [F1], and use [F2] successively with [F1] to pass from $X^{m-1}$ to $X^m$. The empty skeleton also has zero homology.

For $n\ge 1$, the exact sequence of $(X^n,X^{n-1})$ and step 1.1 give an injection $i_n:H_n(X^n)\to C_n$ with image $\ker {\mathrm{\partial }}_n$. The same argument one degree lower makes $i_{n-1}$ injective, including $i_0:H_0(X^0)\to H_0(X^0,\mathrm{\varnothing })$, which is an isomorphism. Since $d_n=i_{n-1}{\mathrm{\partial }}_n$, we obtain $\ker d_n=\ker {\mathrm{\partial }}_n=i_n{H}_n(X^n)$. For $n=0$ this identity follows directly from $i_0$ being an isomorphism and $d_0=0$.

For $(X^{n+1},X^n)$, exactness and $H_n(X^{n+1},X^n)=0$ give the exact tail $C_{n+1}\stackrel{{\mathrm{\partial }}_{n+1}}{\to }{H}_n(X^n)⟶H_n(X^{n+1})⟶0.$ Under the injection $i_n$ of step 2.1, $\mathrm{im}{\mathrm{\partial }}_{n+1}$ corresponds exactly to $\mathrm{im}{d}_{n+1}$ by [F5]. Taking the quotient therefore yields $H_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(X;G)\cong H_n(X^n)\mathrm{/}\mathrm{im}{\mathrm{\partial }}_{n+1}\cong H_n(X^{n+1};G)\mathrm{.}$

By [F3], all inclusions after $X^{n+1}$ induce isomorphisms on $H_n$. Each singular cycle is a finite sum of simplices; [F4] puts the image of each simplex in a finite subcomplex. Their finite union lies in some skeleton, so every class in $H_n(X;G)$ comes from a skeleton. If a skeletal cycle bounds in $X$, its finite bounding chain likewise lies in some later skeleton, where that class already vanishes. Singular chain inclusions are injective, so these are respectively surjectivity and injectivity of the natural colimit map. Thus the colimit is $H_n(X;G)$ without any choice premise. Combining with step 3.1 proves the comparison for arbitrary, possibly infinite-dimensional, CW complexes.

A cellular map preserves all skeleta. Its induced chain maps commute with inclusions, quotient maps and connecting homomorphisms: the latter send a relative cycle represented by $c$ to the class of $\mathrm{\partial }c$, and a chain map commutes with $\mathrm{\partial }$. Consequently every injection, quotient identification and colimit map in steps 2.1--4.1 commutes with cellular maps. The comparison is therefore natural in exactly the stated sense.

Project the three-skeleton connecting map defining $d_n$ to the summand of a fixed $(n-1)$-cell.

For $n\ge 2$, first pass from $X^{n-1}$ to $X^{n-1}\mathrm{/}{X}^{n-2}$ as required by the relative target of the cellular boundary, and then collapse all summand spheres except that of $e_{\beta }^{n-1}$. Equivalently, collapse the entire complement $X^{n-1}\setminus e_{\beta }^{n-1}$, including $X^{n-2}$. Naturality of the connecting homomorphism for the characteristic disk identifies the coefficient with the composite from its oriented boundary sphere to this quotient sphere. This is precisely the reduced-homology endomorphism in Incidence number of two CW cells, so its degree is $[e_{\alpha }^n:e_{\beta }^{n-1}]$.

For $n=1$, use a characteristic interval compatible with the chosen edge generator. Its connecting homomorphism is the singular boundary $[v_{+}]-[v_{-}]$. For each chosen vertex generator $e_v={\epsilon }_v[v]$, the identity $[v]={\epsilon }_v{e}_v$ gives coefficient ${\epsilon }_v({\mathbf{1}}_{\{v_{+}=v\}}-{\mathbf{1}}_{\{v_{-}=v\}})$, exactly the incidence number of Incidence number of two CW cells. This includes negatively chosen zero-cell generators and gives zero when both endpoints coincide. The boundary is an element of a direct sum, so only finitely many coefficients are nonzero; equality of all its projections therefore proves the formula for every $n\ge 1$; separately, $d_0=0$ by definition.

The restrictions $f:(X^n,X^{n-1})\to (Y^n,Y^{n-1})$ induce maps of relative groups, hence maps on cellular chains.

Naturality of pair connecting maps makes these maps commute with the differentials, without a choice assumption. Under the Axiom of Choice, Cellular homology computes singular homology compares cellular and singular homology naturally for arbitrary CW complexes, so the induced maps agree with $f_{\ast }$. For finite $X$ and $Y$, use the finite skeletal exact-sequence calculations in that theorem, followed by finitely many skeletal stabilization isomorphisms: above the last cell dimension each space equals its final skeleton, so no infinite colimit is needed. Every pair and inclusion map in these finite calculations commutes with $f$, giving the same compatibility without choice.

For every $n$, either $C_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ or $C_{n-1}^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ is zero, so $d_n=0$.

Thus cellular homology equals the chain group in each degree; Cellular homology computes singular homology transfers this calculation to singular homology.

Rank-nullity in each degree gives $\mathrm{rank}{C}_n=\mathrm{rank}{H}_n^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}+\mathrm{rank}\mathrm{im}{d}_n+\mathrm{rank}\mathrm{im}{d}_{n+1}$.

Alternating and summing cancels each boundary rank with its neighboring occurrence. The left side is Euler characteristic of a finite CW complex, and Cellular homology computes singular homology identifies the remaining ranks with singular homology.

The cells of $X$ are the disjoint union of the cells of $A$ and those outside $A$, so the first equality follows by splitting the finite alternating sum.

If $A$ is nonempty, $X\mathrm{/}A$ has one new $0$-cell (the quotient point) and exactly the cells outside $A$ otherwise; its alternating count gives the second formula.