been playing with some math notation recently
a function f is surjective iff for every y ∈ cod(f) we have

turns out there is a pretty natural way modify this notation to support ∀∃-statements instead of just ∀-statements. using it, topological continuity looks like this:
(edit: this is not the definition of continuity, oops)
one can read this as follows: "if you can give me the solid stuff, then i can give you the dotted stuff". give me a thing which maps under f to an element of an open set, and in response i can tell you that the preimage of that open set is also open and contains your original thing
precisely, filled bullets correspond to universal quantifiers, open circles to existential quantifiers, solid arrows to relations on the left of an implication, and dashed arrows to relations on the right of an implication. the ∀∃-sentence corresponding to the diagram is:
( ∀ solid bullets ∃ open circles ) ( IF solid-arrow stuff holds THEN dashed-arrow stuff holds )
to be clear: this means that solid bullets have switched roles from the previous notation: where before they are existentials, now they are universals
also I should address the star (⋆): a star represents a value which is uniquely induced by a ↦-relation. it gets its own symbol because even though technically it corresponds to a quantifier, knowing the value of a star gives you no new information. stars can be "ignored" when interpreting the quantification of a diagram
something neat about the above diagram is that it defines topological continuity with no "extra" variables: it says that f is continuous wrt τ, σ iff ... and then introduces no new variables
