\left( \coprod_{d_1+d_2 \leq d} \mcal{C}_{q,d_1}(Y)(\C)\times \mcal{C}_{q,d_2}(Y)(\C) \right )/ \sim \subseteq \mcal{Z}_{q}(Y).$$ By [@Li:completion] each $\mcal{Z}_{q,\leq d}(Y)$ is a closed, compact Hausdorff space and $\mcal{Z}_{q}(Y)$ has the weak topology with respect to this filtration.

When $U$ is quasi-projective with projectivization $U\subseteq \overline{U}$ then define $\mcal{Z}_{q}(U) = \mcal{Z}_{q}(\overline{U})/\mcal{Z}_{q}(U_{\infty})$ where $U_{\infty} = \overline{U}\backslash U$. The images of $\mcal{Z}_{q,\leq k}(\overline{U})$ in the quotient $\mcal{Z}_{q}(U)$ give a filtration by compact subspaces (and $\mcal{Z}_{q}(U)$ has the weak topology with respect to this filtration). This definition is independent of choice of projectivization [@LF:qproj], [@FG:cyc].

Cycle-spaces over $\R$
----------------------

Suppose that $Z$ is a real variety. Write the set of $\R$-points equipped with the analytic topology as $Z(\R)^{an}$ or simply $Z(\R)$ with the topology understood.

Let $Y$ be a projective real variety. Consider the topological monoid $\mcal{C}_q(Y)(\R)$. As explained in the proof of [@FW:real Proposition 8.2] (see Proposition \[gtract\]), the topological monoid $\mcal{C}_q(Y)(\R)$ is tractable and therefore its naive group completion is a homotopy group completion. Write $$\mcal{Z}_{q}(Y) = (\mcal{C}_{q}(Y)(\R))^{+}$$ for the naive group completion.

Suppose that $U$ is a quasi-projective real variety with projectivization $U\subseteq \overline{U}$. Define the topological group of $q$-cycles on the quasi-projective real variety $U$ to be $$\mcal{Z}_{r}(U) = \mcal{Z}_{r}(\overline{U})/\mcal{Z}_{r}(\overline{U}\backslash U).$$

If $X$ is 