The category of topological spaces is the category of models of a relational universal strict Horn theory $T$ without equality, i.e. the axioms are of the form $(\forall x)(\phi(x) \rightarrow \psi(x))$ where $x$ is a set of variables and $\phi$ and $\psi$ are conjunctions of atomic formulas without equality. The result was proved for topological spaces first by E. Manes in Equational theory of ultrafilter convergence, Alg. Univ. 11 (1980), 163-172 and generalized by J. Rosický to any fibre-small topological category in Concrete categories and infinitary languages, J. Pure Appl. Alg 22 (1981), 309-339. And indeed, the class of axioms is a proper class, otherwise the category of topological spaces would be locally presentable, which is known to be false. Indeed, no non-discrete space is presentable. It is your large sketch.

The category of topological spaces is the category of models of a relational universal strict Horn theory $T$ without equality, i.e. the axioms are of the form $(\forall x)(\phi(x) \rightarrow \psi(x))$ where $x$ is a set of variables and $\phi$ and $\psi$ are conjunctions of atomic formulas without equality. The result was proved for topological spaces first by E. Manes in Equational theory of ultrafilter convergence, Alg. Univ. 11 (1980), 163-172 and generalized by J. Rosický to any fibre-small topological category in Concrete categories and infinitary languages, J. Pure Appl. Alg 22 (1981), 309-339. And indeed, the class of axioms is a proper class, otherwise the category of topological spaces would be locally presentable, which is known to be false. Indeed, no non-discrete space is presentable.