$\mathbf{Top}$ has no small dense subcategory Recall that a subcategory $\mathcal{B} \subseteq \mathcal{A}$ is called dense when $\mathcal{A} \to \mathrm{Hom}(\mathcal{B}^{\mathrm{op}},\mathbf{Set})$, $X \mapsto \mathrm{Hom}(-,X)|_{\mathcal{B}}$ is fully faithful, or equivalently, for every $A \in \mathcal{A}$ we have the coend expression $A = \int^{B \in \mathcal{B}} \mathrm{Hom}(B,A) \otimes B$. Since I am not able to see a small dense subcategory of $\mathbf{Top}$, the category of topological spaces and continuous maps, I assume that there is none (in contrast to nice subcategories such as the subcategory of CW-complexes).

Why does $\mathbf{Top}$ not have a small dense subcategory?

Perhaps one can derive a contradiction why using ordinal numbers (equipped with the order topology)? Another path would be to use that every cocomplete category with a small dense subcategory actually satisfies a very strong adjoint functor theorem: every cocontinuous functor on it is a left adjoint. So it suffices to construct a cocontinuous functor on $\mathbf{Top}$ which is no left adjoint.
Maybe we can even prove something stronger? Recall that $\mathcal{B} \subseteq \mathcal{A}$ is colimit-dense when every object $A \in \mathcal{A}$ has a diagram in $\mathcal{B}$ whose colimit in $\mathcal{A}$ is $A$. Every dense subcategory is colimit-dense, so the following is actually stronger:

Why does $\mathbf{Top}$ not have a small colimit-dense subcategory?

 A: Let $\kappa$ be an infinite cardinal greater than the cardinalities of all the objects of your subcategory.  Let $X$ be $\kappa$ with the topology that a nonempty set is open iff its complement has cardinality less than $\kappa$.  Note that this topology is discrete when restricted to every subset of cardinality less than $\kappa$.  So, the continuous maps from objects of your subcategory to $X$ are the same as they would be if $X$ had the discrete topology.  It follows that $X$ cannot be a colimit of any diagram in your subcategory (the colimiting cone would lift to $X$ with the discrete topology, but the identity function from $X$ to $X$ with the discrete topology is not continuous).
A: Here's a variant of Eric's proof.
The colimit-closure of the spaces of cardinality $<\kappa$ is called the category of $\kappa$-tight -spaces (I might be off by a $(-)^+$). The $\kappa$-tight spaces are those spaces $X$ such that a map $X \to Y$ is continuous iff its restriction to any $\kappa$-small subspace of $X$ is continuous (analogous to compactly-generated spaces).
An example of a space which is not $\kappa$-tight is the Stone-Cech compactification $\beta S$ of a set $S$ of cardinality $\geq \kappa$. For any small set of spaces, the largest full subcategory of $Top$ in which they are dense is contained in the their colimit closure, and hence is contained in the $\kappa$-tight spaces for some $\kappa$, and so is not all of $Top$.
