or $\R$. Write $\Mor{k}{X}{Y}$ for the set of continuous algebraic maps between $X$ and $Y$. When $X$ is semi-normal then $\Mor{k}{X}{Y} = \Hom{}{X}{Y}$. Friedlander-Walker construct “analytic” topologies on $\Mor{k}{X}{Y}$ in [@FW:funcspc] for $k=\C$ and in [@FW:real] for $k=\R$. The set of continuous algebraic maps with this topology will be written $\Mor{k}{X}{Y}^{an}$. By [@FW:real Lemma 1.2] $\Mor{\R}{X}{Y}^{an} = (\Mor{\C}{X_{\C}}{Y_{\C}}^{an})^{G}.$

When $X$,$Y$ are projective real varieties and $W$,$Z$ are projective complex varieties then this topology coincides with the subspace topology induced by the inclusions $$\Mor{\C}{W}{Z}  \subseteq \map{W(\C)}{Z(\C)}$$ and $$\Mor{\R}{X}{Y}  \subseteq \map{X_{\C}(\C)}{Y_{\C}(\C)}^{G},$$ where $\map{-}{-}$ denotes the space of continuous maps is with compact-open topology. When the domain is only quasi-projective then the analytic topology on the algebraic mapping spaces is no longer the compact-open topology but rather the topology of convergence with bounded degree (see [@FL:dual Appendix A]).

Let $W$ be a quasi-projective complex variety and $Z$ be a projective complex variety. Write $d= \dim W$. Let $\mcal{C}_{r}(Z)(W)$ denote the monoid of effective cycles on $W\times Z$ equidimensional or relative dimension $r$ on $W$. This is made into a topological monoid via the subspace topology induced by the inclusion $$\mcal{C}_{r}(Z)(W) \subseteq \mcal{C}_{d+r}(W \times Z) \eqdef \frac{\mcal{C}_{d+r}(\overline{W}\times Z)}{\mcal{C}_{d+r}(W_{\infty}\times Z)}$$ where $W\subseteq \overline{W}$ a projective 