Field cohomology of an infinite wedge of circles

This CW structure has one zero-cell $v$, one oriented one-cell $e_j$ for each $j\ge 1$, and no higher cells. Every edge has terminal and initial endpoint $v$, so its cellular boundary is $v-v=0$ by [F1]. With coefficients in $k$ the same endpoint calculation gives $C_0^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=k$, $C_1^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}={⨁}_{j\ge 1}k{e}_j$ and zero differential. Thus its homology is $k$ in degree zero, the displayed direct sum in degree one, and zero in higher degrees. The arbitrary-CW comparison of [F1] transfers these actual groups to singular homology. The prescribed weak topology is essential to that CW application.

For each sequence $a=(a_j)\in {\prod }_{j\ge 1}k$, define ${\lambda }_a({\sum }_j{c}_j{e}_j)={\sum }_j{c}_j{a}_j$. Every input sum has finite support, so this is a well-defined linear functional with no restriction on the support of $a$. Conversely a linear functional $\lambda$ is determined by the sequence $a_j=\lambda (e_j)$; linearity gives $\lambda ={\lambda }_a$. These maps are inverse and linear. Applying [F2] and step 1.1 yields the claimed $H^1$ product. The dual of $k$ is $k$ by evaluation at $1$, and the dual of zero is zero, giving the degree-zero and higher-degree cohomology groups.

For $k={\mathbb{F}}_2$, a finite-support sequence $c$ maps injectively to the nonnegative integer ${\sum }_{j\ge 1}{c}_j{2}^{j-1}$ by uniqueness of binary expansion, and every such integer comes from its finite binary expansion. Thus the direct sum is countable. The full product cannot be enumerated as $a^{(1)},a^{(2)},\dots$: the sequence $b_j=1-a_j^{(j)}$ differs from the $j$th proposed sequence at coordinate $j$. It is therefore uncountable and cannot be isomorphic to the direct sum. The all-one sequence also explicitly represents a functional outside the canonical finite-support subgroup: it sends every $e_j$ to $1$.

The zero sequence corresponds to the zero functional; a sequence supported at one coordinate is evaluation of that coordinate. The single vertex gives $H^0=k$ even though there are infinitely many edges. There is no degree-two cell, so no unmentioned differential changes the degree-one direct sum. The diagonal argument in step 3.1 uses the explicitly given enumeration indices and requires no choices; AC is inherited from field duality in [F2] and is also available for the arbitrary-CW comparison's compactness argument. This is the CW wedge, whose topology is not being replaced by a shrinking-circle subspace topology.