Topological spaces are not algebraic structures (in the sense of monads). Indeed, in the presence of the axiom of choice, one can show that any category that is monadic over $\textbf{Set}$ is a regular category, but $\textbf{Top}$ is known to be not regular. However, the category $\textbf{Frm}$ of frames is monadic over $\textbf{Set}$, so the theory of locales (= pointless topological spaces) is, in some sense, coalgebraic.
There is also a category-theoretic notion of topological structure that goes something like this.
Definition. Let $\Gamma : \mathcal{C} \to \mathcal{S}$ be a functor and let $B : \mathcal{J} \to \mathcal{C}$ be a (possibly large) diagram. A $\Gamma$-initial cone is a cone $\alpha$ from an object $A$ to the diagram $B$ such that, for all cones $\beta$ from $\Gamma C$ to $\Gamma B$, there exists a unique morphism $f : C \to A$ in $\mathcal{C}$ such that $\Gamma (\alpha_j \circ f) = \beta_j$ for all $j$ in $\mathcal{J}$.
A topologising fibration (or topological functor in the sense of [Adámek, Herrlich, and Streicher]) is a functor $\Gamma : \mathcal{C} \to \mathcal{S}$ such that, for every (possibly large) discrete diagram $B : \mathcal{J} \to \mathcal{C}$ and every cone $\beta$ from an object $X$ to the diagram $\Gamma B$, there exists a $\Gamma$-initial cone $\alpha$ such that $\beta = \Gamma \alpha$.
In effect, what we are axiomatising is the existence of initial topologies. This definition already allows to prove many things: for example, any topologising fibration is faithful, has both left and right adjoints, and is a Grothendieck fibration. Moreover, the definition is self-dual: $\Gamma : \mathcal{C} \to \mathcal{S}$ is a topologising fibration if and only if $\Gamma^\textrm{op} : \mathcal{C}^\textrm{op} \to \mathcal{S}^\textrm{op}$ is a topologising fibration, i.e. we also get $\Gamma$-terminal lifts of cocones!
