TITLE: Does the lattice of all topologies embed into the lattice of $T_1$-topologies?
QUESTION [9 upvotes]: Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T_1}(\kappa)$ be the lattice of all $T_1$-topologies on $\kappa$.
Is there an injective lattice homomorphism $\varphi: \text{Top}(\kappa)\to \text{Top}^{T_1}(\kappa)$?

REPLY [5 votes]: This is a long comment about Bjørn's answer rather than an answer itself.
Hartmanis proves
Thm 1 If $\kappa>2$, then $\text{Top}(\kappa)$ is a simple lattice.
Thm 2 If $\kappa$ is infinite, then 
(1) For any finite subset $F\subseteq \kappa$, the restriction map 
$\rho_F:\tau\mapsto \tau|_{\kappa-F}$ is a nonconstant, noninjective, complete lattice homomorphism from 
$\text{Top}^{T_1}(\kappa)$ to $\text{Top}^{T_1}(\kappa-F)\;(\cong\text{Top}^{T_1}(\kappa))$. 
(2) For any finite subset $F\subseteq \kappa$, $\ker(\rho_F)$ is a proper, complete congruence on $\text{Top}^{T_1}(\kappa)$. Every proper, complete congruence has this form. If $F\neq G$ are distinct finite subsets of $\kappa$, then $\ker(\rho_F)\neq\ker(\rho_G)$. \\\
Theorem 1 implies that if $\varphi:\text{Top}(\kappa)\to\text{Top}^{T_1}(\kappa)$ is an embedding, then for any congruence $\theta$ on $\text{Top}^{T_1}(\kappa)$ one must have either 
Case 1. the image of $\varphi$ is contained in a single $\theta$-class, or
Case 2. the image of $\varphi$ is contained in a $\theta$-transversal.
Otherwise $\varphi^{-1}(\theta)$ would be a nontrivial proper congruence of a simple lattice. Thus each congruence $\theta$ on $\text{Top}^{T_1}(\kappa)$ restricts the possibilities for $\varphi$.
Theorem 2 implies that there are lots of $\theta$'s to choose from, namely all of the form $\ker(\rho_F)$. 
Thms 1&2 imply that there are lots of restrictions $\varphi$ must satisfy, but there are not enough restrictions to rule out the existence of $\varphi$. In my answer on this page I construct $\varphi$ whose image is contained in a $\theta$-transversal for every $\theta$ of the form $\ker(\rho_F)$.