Given a sequence of topological spaces $\{Y_{k}\}_{k\geq1},$ the homological stability is a property that, there is a function $g:\,\mathbb{N}_{0}\rightarrow \mathbb{N}_{0}$ such that for any $i\geq0$ and for any $k\geq g(i)$ we have $$H_{i}(Y_{k})\cong H_{i}(Y_{k+1})$$ we call $H_{i}(Y_{k}),$ for $k\geq g(i),$ the stable homology group of $\{Y_{k}\},$ and $g(i)$ the stable range. This implies that the relative homology group $H_{i}(Y_{k+1},Y_{k})$ is zero for $k\geq g(i).$
Question: If it is given that $Y_{k}\subset Y_{k+1}$ and the relative homology group $H_{i}(Y_{k+1},Y_{k})$ is zero for $k\geq g(i).$ Then is it true or not that the sequence of topological spaces $\{Y_{k}\}_{k\geq1}$ is homological stable?
