duality map $$\mcal{D}: \Mor{\C}{W}{\mcal{C}_0(Z)}^{+} \xrightarrow{\Gamma} \mcal{Z}_{0}(W)(Z) \subseteq \mcal{Z}_{d}(W\times Z).$$ While this is a continuous injective homomorphism it is not a topological embedding (see [@FL:dual]).

(c.f. [@Teh:real Proposition 2.9])

1.  If $W$ is a normal quasi-projective complex variety and $Z _1\subset Z _2$ is a closed subvariety of a complex projective variety then $$\Mor{\C}{W}{\mcal{C}_{0}(Z_{1})}^{+}\subseteq \Mor{\C}{W}{\mcal{C}_{0}(Z_{2})}^{+}$$ is a closed subspace.

2.  If $U$ is a normal quasi-projective real variety and $Y\subseteq Z$ is a closed subvariety of a projective real variety then $$\Mor{\R}{U}{\mcal{C}_{0}(Y)}^{+}\subseteq \Mor{\R}{U}{\mcal{C}_{0}(Z)}^{+}$$ is a closed subspace.

For the first statement it is equivalent to show that $\mcal{Z}_{r}(Z_{1})(W) \subseteq \mcal{Z}_{r}(Z_{2})(W)$ is closed. Using Lemma \[qmcl\] we see that $\mcal{C} _{r+d}(W\times Z _1)\subset \mcal{C} _{r+d}(W\times Z _2)$ is closed and since $\mcal{C}_{r}(Z_1)(W) = \mcal{C} _{r+d}(W\times Z _1)\cap \mcal{C}_{r}(Z_{2})(W)$ we conclude that $\mcal{C}_{r}(Z_1)(W)\subset\mcal{C}_{r}(Z_{2})(W)$ is a closed subspace. Write $\pi:\mcal{C}_{r}(Z_{2})(W)^{\times 2} \to \mcal{Z}_{r}(Z_{2})(W)$ for the quotient. Then $\pi^{-1}\mcal{Z}_{r}(Z_{1})(W) =\mcal{C}_{r}(Z_{1})(W)^{\times 2} + \Delta(\mcal{C}_{r}(Z_{2})(W)) $ (where $\Delta$ denotes the diagonal) is closed by step (1) in Proposition \[Gcocomm\]. Therefore $\mcal{Z}_{r}(Z_{1})(W) \subseteq \mcal{Z}_{r}(Z_{2})(W)$ is closed.

The second statement follows immediately from the first statement together with Proposition \[appcycagr\] and [@FW:real Lemma 1.2].

1.  Let $W$ be a quasi-projective complex variety. The space of algebraic $q$-cocyles is defined 