gn.general topology - Lebesgue dimension of closures satisfying the Z-set condition

No for general topological spaces, yes for metrizable ones (and I believe the argument can be generalized to all normal spaces).

Bad example: $X={a,b,c}$ with open sets $emptyset$, $X$, ${a}$, ${a,b}$, ${a,c}$. Let $A={a}$, then $bar A=X$. The homotopy is given by $H_1=id$, $H_tequiv a$ for $t<1$. The dimension of $A$ is 0 but the dimension of $bar A$ is 1.

On the positive side, let me begin with a quick and dirty proof in the case when $bar A$ is a compact metric space. Let ${U_i}$ be an open covering of $bar A$. We need to find a refined covering of multiplicity at most $N+1$ where $N=dim A$. It suffices to find a continuous map $f:bar Ato A$ and an open covering ${V_j}$ of $A$ such that ${f^{-1}(V_j)}$ is a refinement of ${U_i}$. Indeed, in this case we can find a refinement of ${V_j}$ of multiplicity at most $N+1$ and its $f$-preimage is the desired refinement of ${U_i}$.

In the compact case, let ${V_j}$ be the covering by $(rho/3)$-balls where $rho$ is the Lebesgue number of the covering ${U_i}$. Then, for some $t$ sufficiently close to 1, the map $f=H_t$ satisfies the desired property: the preimage of every $V_j$ has diameter less than $rho$ and hence is contained in some of the sets $U_i$. Indeed, suppose the contrary. Then there is a sequence $t_kto 1$ and sequences $x_k,y_kin bar A$ such that $|x_ky_k|gerho$ but $|H_{t_k}(x_k)H_{t_k}(y_k)|<2rho/3$. Due to compactness we may assume that $x_k$ and $y_k$ converge to some $x,yinbar A$. Then $|xy|gerho$ but $|H_1(x)H_1(y)|le 2rho/3$, a contradiction.

In the general metric space case, let $rho(x)$ denote the local Lebesgue number of ${U_i}$ at $x$, that is the supremum of $rho$ such that the ball $B_rho(x)$ is contained in one of the set $U_i$. Note that $xmapstorho(x)$ is a positive 1-Lipschitz function on $bar A$. It is easy to construct a continuous function $u:bar Ato[0,1)$ such that the distance from $x$ to $H_t(x)$ is less that $rho(x)/10$ for all $xinbar A$ and all $t>u(x)$. Then the map $f:bar Ato A$ given by $f(x)=H_{u(x)}(x)$ and the covering of $A$ by the balls of the form $B_{rho(x)/10}(x)$, $xin A$, will do the job.