Theorem (suggested by I.V.Protasov). Every solvable Hausdorff topological group $G$ is topologically solvable in the sense that $G$ contains an increasing sequence of closed subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

Proof: The group $G$, being solvable, contains an increasing sequence of subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

For every $i\le n$ let $\bar G_i$ be the closure of the group $G_i$ in $G$. It can be shown that for every $i\le n$ the normality of the subgroup $G_{i-1}$ in $G_i$ implies the normality of the its closure $\bar G_{i-1}$ in the closure $\bar G_i$ of $G_i$. The quotient group $\bar G_i/\bar G_{i-1}$ is abelian since is contains a dense abelian group $G_i/(G_i\cap\bar G_{i-1})$ (which is a homomorphic image of the abelian group $G_i/G_{i-1}$). So, $G$ is topologically solvable.

Corollary 1. Each non-abelian solvable Hausdorff topological group contains a proper non-trivial closed normal subgroup and hence cannot be topologically simple.

The following corollary answer the question of @iranano.

Corollary 2. Each solvable Hausdorff topological group $(G,\tau)$ of cardinality $|G|>\mathfrak c$ admits a non-trivial (not necessarily Hausdorff) group topology $\sigma\subsetneq \tau$.

Proof: If $G$ is not abelian, then by Corollary 1, $G$ contains a proper non-trivial normal subgroup $H$ and then $\sigma:=\{UH:U\in\tau\}$ is the required weak (non-Hausdorff) group topology on $G$.

So, we assume that $G$ is abelian. If $G$ is not precompact, then by the famous Prodanov-Stoyanov Theorem, $G$ is not minimal and hence admits a strictly weaker Hausdorff group topology $\sigma\subsetneq \tau$.

It remains to consider the case of a precompact abelian group $G$. It follows that $G$ is a dense subgroup of an abelian compact Hausdorff group $\bar G$. Since characters separated points of $\bar G$, there exists a non-trivial continuous homomorphism $\chi:G\to\mathbb T=\{z\in \mathbb C:|z|=1\}$ into the circle group. Since $|G|>\mathfrak c=|\mathbb T|$ the kernel $H=\chi^{-1}(1)$ is a non-trivial closed proper subgroup of $G$. Then $\sigma:=\{UH:U\in\tau\}$ is the required weak (non-Hausdorff) group topology on $G$.

Theorem (suggested by I.V.Protasov). Every solvable Hausdorff topological group $G$ is topologically solvable in the sense that $G$ contains an increasing sequence of closed subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

Proof: The group $G$, being solvable, contains an increasing sequence of subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

For every $i\le n$ let $\bar G_i$ be the closure of the group $G_i$ in $G$. It can be shown that for every $i\le n$ the normality of the subgroup $G_{i-1}$ in $G_i$ implies the normality of the its closure $\bar G_{i-1}$ in the closure $\bar G_i$ of $G_i$. The quotient group $\bar G_i/\bar G_{i-1}$ is abelian since is contains a dense abelian group $G_i/(G_i\cap\bar G_{i-1})$ (which is a homomorphic image of the abelian group $G_i/G_{i-1}$). So, $G$ is topologically solvable.

Corollary 1. Each non-abelian solvable Hausdorff topological group contains a proper non-trivial closed normal subgroup and hence cannot be topologically simple.

The following corollary answer the question of @iranano.

Corollary 2. Each solvable Hausdorff topological group $(G,\tau)$ of cardinality $|G|>\mathfrak c$ admits a non-trivial (not necessarily Hausdorff) group topology $\sigma\subsetneq \tau$.

Proof: If $G$ is not abelian, then by Corollary 1, $G$ contains a proper non-trivial normal subgroup $H$ and then $\sigma:=\{UH:U\in\tau\}$ is the required weak (non-Hausdorff) group topology on $G$.

So, we assume that $G$ is abelian. If $G$ is not precompact, then by the famous Prodanov-Stoyanov Theorem, $G$ is not minimal and hence admits a strictly weaker Hausdorff group topology $\sigma\subsetneq \tau$.

It remains to consider the case of a precompact abelian group $G$. In this case the completion $\bar G$ of $G$ by the two-sided uniformity is compact, so characters separate points of $\bar G$. Consequently, there exists a non-trivial continuous homomorphism $\chi:G\to\mathbb T=\{z\in \mathbb C:|z|=1\}$ into the circle group. Since $|G|>\mathfrak c=|\mathbb T|$ the kernel $H=\chi^{-1}(1)$ is a non-trivial closed proper subgroup of $G$. Then $\sigma:=\{UH:U\in\tau\}$ is the required weak (non-Hausdorff) group topology on $G$.

Theorem (suggested by I.V.Protasov). Every solvable Hausdorff topological group $G$ is topologically solvable in the sense that $G$ contains an increasing sequence of closed subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

Proof: The group $G$, being solvable, contains an increasing sequence of subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

For every $i\le n$ let $\bar G_i$ be the closure of the group $G_i$ in $G$. It can be shown that for every $i\le n$ the normality of the subgroup $G_{i-1}$ in $G_i$ implies the normality of the its closure $\bar G_{i-1}$ in the closure $\bar G_i$ of $G_i$. The quotient group $\bar G_i/\bar G_{i-1}$ is abelian since is contains a dense abelian group $G_i/(G_i\cap\bar G_{i-1})$ (which is a homomorphic image of the abelian group $G_i/G_{i-1}$). So, $G$ is topologically solvable.

Corollary. Each non-abelian solvable Hausdorff topological group contains a non-trivial closed normal subgroup and hence cannot be topologically simple.

Theorem (suggested by I.V.Protasov). Every solvable Hausdorff topological group $G$ is topologically solvable in the sense that $G$ contains an increasing sequence of closed subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

Proof: The group $G$, being solvable, contains an increasing sequence of subgroups $\{1\}=G_0\subset G_1\subset\dots\subset G_n=G$ such that for every $i\le n$ the subgroup $G_{i-1}$ is normal in $G_i$ and the quotient group $G_i/G_{i-1}$ is abelian.

For every $i\le n$ let $\bar G_i$ be the closure of the group $G_i$ in $G$. It can be shown that for every $i\le n$ the normality of the subgroup $G_{i-1}$ in $G_i$ implies the normality of the its closure $\bar G_{i-1}$ in the closure $\bar G_i$ of $G_i$. The quotient group $\bar G_i/\bar G_{i-1}$ is abelian since is contains a dense abelian group $G_i/(G_i\cap\bar G_{i-1})$ (which is a homomorphic image of the abelian group $G_i/G_{i-1}$). So, $G$ is topologically solvable.

Corollary 1. Each non-abelian solvable Hausdorff topological group contains a proper non-trivial closed normal subgroup and hence cannot be topologically simple.

The following corollary answer the question of @iranano.

Corollary 2. Each solvable Hausdorff topological group $(G,\tau)$ of cardinality $|G|>\mathfrak c$ admits a non-trivial (not necessarily Hausdorff) group topology $\sigma\subsetneq \tau$.

Proof: If $G$ is not abelian, then by Corollary 1, $G$ contains a proper non-trivial normal subgroup $H$ and then $\sigma:=\{UH:U\in\tau\}$ is the required weak (non-Hausdorff) group topology on $G$.

So, we assume that $G$ is abelian. If $G$ is not precompact, then by the famous Prodanov-Stoyanov Theorem, $G$ is not minimal and hence admits a strictly weaker Hausdorff group topology $\sigma\subsetneq \tau$.

It remains to consider the case of a precompact abelian group $G$. It follows that $G$ is a dense subgroup of an abelian compact Hausdorff group $\bar G$. Since characters separated points of $\bar G$, there exists a non-trivial continuous homomorphism $\chi:G\to\mathbb T=\{z\in \mathbb C:|z|=1\}$ into the circle group. Since $|G|>\mathfrak c=|\mathbb T|$ the kernel $H=\chi^{-1}(1)$ is a non-trivial closed proper subgroup of $G$. Then $\sigma:=\{UH:U\in\tau\}$ is the required weak (non-Hausdorff) group topology on $G$.