Subcategories of T1, Hausdorff and TD spaces are reflective subcategories of Category *Top*. How do I construct a T1, Hausdorff and TD reflections  to show that subcategories of T1, Hausdorff and TD spaces are reflective subcategories of Category Top?
The T0 reflection is just the T0 quotient of a space X. What about the others?

@Henno .
This is what I tried for the T1 Reflection. But I hardly used the fact that singleton sets of a T1 space are closed. Any suggestion?
Let $R$ be the smallest equivalence relation $S$ on $X$ such that $X/S$ is closed. Denote by $X/ R$ the quotient space and $p$ the quotient map from $X$ onto $X/ R$. We show that $X /R$ is the $T_{1}$ reflection of $X$. That $X/R$ is $T_{1}$ follows from the fact that a quotient space is $T_{1}$ if and only if each equivalence class of $X$ is closed, [41]. Let $f:X\longrightarrow Y$ be a continuous map from $X$ to a $T_{1}$ space $Y$. Define a function $g:X/R\longrightarrow Y$ by $[x]\mapsto f(x)$. Observe that $g$ is defined since for any $[x], [y]\in X/R$, if $g([x])\neq g([y])$, then $f(x)\neq f(y)$. Now, $\{f(x)\}$ and $\{f(y)\}$ are disjoint closed subsets of $Y$. Therefore, $f^{-1}(\{x\})$ and $f^{-1}(\{y\})$ are disjoint closed sets of $X$. Because $[x]\subseteq f^{-1}(\{f(x)\})$ and $[y]\subseteq f^{-1}(\{f(y)\})$, it follows that $[x]\neq [y]$.\
         For continuity of $g$, let $U$ be a closed subset of $Y$. Then $f^{-1}(U)$ is closed in $X$. Observe that 
         \begin{align*}
    g^{-1}(U)&=\{[x]:f(x)\in U\}\\
    &=\{[x]:x\in f^{-1}(U)\}\\
    &=p(f^{-1}(U))
   \end{align*}
        Now, 
        \begin{align*}
   p^{-1}(p(f^{-1}(U)))&=\{x: f(x)\in p(f^{-1}(U))\}\\
   &=\{x:x\in f^{-1}(U)\}\\
   &=f^{-1}(U)\subseteq f^{-1}(U)\subseteq p^{-1}(p(f^{-1}(U)))
  \end{align*}
        Therefore, $p^{-1}(p(f^{-1}(U)))=f^{-1}(U)$. By definition of a quotient map $p$, $g^{-1}(U)$ is closed in $X/R$. Thus $g$ is continuous.\
        Clearly $g$ is unique.\
        So, the $T_{1}$ quotient space $X/R$ is the $T_{1}$ reflection of $X$.
