Limit T-spaces

Let $F$ be a field of prime characteristic $p$ and let $\mathbf V_p$ be the variety of associative algebras over $F$ without unity defined by the identities $[[x,y],z]=0$ and $x^4=0$ if $p=2$ and by the identities $[[x,y],z]=0$ and $x^p=0$ if $p>2$ (here $[x,y]=xy-yx$). Let $A/V_p$ be the free algebra of countable rank of the variety $\mathbf V_p$ and let $S$ be the T-space in $A/V_p$ generated by $x_1^2x_2^2\dots x_k^2+V_2$, where $k\in\mathbb N$ if $p=2$ and by $x_1^{\alpha_1}x_2^{\alpha_2}[x_1,x_2]\dots x_{2k?1}^{\alpha_{2k-1}}x_{2k}^{\alpha_{2k}}[x_{2k?1},x_{2k}]+V_p$, where $k\in\mathbb N$ and $\alpha_1,\dots,\alpha_{2k}\in\{0,p-1\}$ if $p>2$. As is known, $S$ is not finitely generated as a T-space. In the present paper, we prove that $S$ is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative $F$-algebra without unity of countable rank.