Universal space for all topological spaces This problem is from General Topology, revised and complete edition, Engelking. 2.3.I.
Let $E$ be the topological space, $E=\{0,1,2\}$ with the topology consisting of the empty set, the set $\{0\}$, and the whole space. Show that the space $E^m$ is universal for all topological spaces of weight $m \ge \aleph_0$ (Aleph null) and cardinality less than or equal to $2^m$. Note that every topological space is homeomorphic to a subspace of a power of $E$.
 A: The idea is to use the embedding theorem 2.3.20.
We have a base $\mathcal{B}$ of size at most $\mathbf{m}$, and for each $B$ in that base define $f_B: X \to E$ by $f_B(x)=0$ for $x \in B$ and $f_B(x)=1$ for $x \notin B$. This is continuous as $f_B[\{0\}]= B$ is open, and $\{0\}$ is the only non-trivial open set of $E$. Note that $\{f_B: B \in \mathcal{B}\}$ separates points and closed sets, because $\mathcal{B}$ is a base.
We would be done if $X$ were a $T_0$ space, because then these functions would also separate points (2.3.26 states this fact).
To get an embedding in the general we need to add functions $f$ from $X$ to $E$ that are continuous and separate points for the all other points as well. That's why we have two extra points besides $0$ (otherwise Sierpiński space would have sufficed, again 2.3.26): any function $f$ on $X$ that only takes values $1,2 \in E$ is always continuous, as $f^{-1}[\{0\}]=\emptyset$ etc.
Finally, simply by set theory and as $|X| \le 2^{\mathbf{m}}$, we can find an injection $i: X \to \{1,2\}^{\textbf{m}}$ (this does use AC, but that's assumed throughout in Engelking's text). Then add all functions $\pi_\alpha \circ i: X \to E, \alpha \in \mathbf{m}$, which are continuous by the remark above, to the $\{f_B: B \in \mathcal{B}\}$ to get a family  of size $\mathbf{m} + \mathbf{m} = \mathbf{m}$ many continuous functions that separates points and closed sets, and also separates points.
Then 2.3.20 finishes the job.
