Weakly Hausdorff k-spaces are the quasiseparated objects in the category of condensed sets (see Thm 2.16 here). Quasiseparated is a categorical notion which is standard in topos theory.

In turn, the category of condensed sets can be defined as

• sheaves on compact Hausdorff spaces (= algebras for the ultrafilter monad on $Set$ = opposite category of commutative unital $C^\ast$ algebras), or as
• sheaves on totally disconnected compact Hausdorff spaces (= profinite sets), or as
• sheaves on extremally disconnected spaces (=idempotent completion of the Kleisli category for the ultrafilter monad), or as
• sheaves on free extremally disconnected spaces (=the Kleisli category for the ultrafilter monad).

Here, the ultrafilter monad is the unique monad whose underlying functor is the ultrafilter functor $\beta : Set \to Set$, carrying a set to the set of ultrafilters thereon. The topology with respect to which we take sheaves is some additional data in the first two descriptions, but in the last two descriptions it's very straightforward: a sheaf is a presheaf carrying coproducts to products.

Anyway, that gives a few possible definitions which don't require you to know what a topological space is.

Sorry, I misremembered this. There are more quasiseparated condensed sets than just the weak Hausdorff $k$-spaces. See p. 15, right after Prop 2.9 here. I don't know if there's a way to define weak Hausdorff $k$-spaces in terms of condensed sets alone.

Weakly Hausdorff k-spaces are the quasiseparated objects in the category of condensed sets (see Thm 2.16 here). Quasiseparated is a categorical notion which is standard in topos theory.

In turn, the category of condensed sets can be defined as

• sheaves on compact Hausdorff spaces (= algebras for the ultrafilter monad on $Set$ = opposite category of commutative unital $C^\ast$ algebras), or as
• sheaves on totally disconnected compact Hausdorff spaces (= profinite sets), or as
• sheaves on extremally disconnected spaces (=idempotent completion of the Kleisli category for the ultrafilter monad), or as
• sheaves on free extremally disconnected spaces (=the Kleisli category for the ultrafilter monad).

Here, the ultrafilter monad is the unique monad whose underlying functor is the ultrafilter functor $\beta : Set \to Set$, carrying a set to the set of ultrafilters thereon. The topology with respect to which we take sheaves is some additional data in the first two descriptions, but in the last two descriptions it's very straightforward: a sheaf is a presheaf carrying coproducts to products.

Anyway, that gives a few possible definitions which don't require you to know what a topological space is.