Closed Formulas in a First Order Logic Theory

Question

According to wiki a theory (im instered in FOL) is

is a set of sentences in a formal language. Usually a deductive system is understood from context

It's clearly specificied that we are considering only sentences ( = closed formulas) but we can't build a theory using open formulas. But why ???

Answer 1

We can, but open axioms must be read as universally quantified.

See the well-known example of Tarski's axioms for (elementary) geometry, formulathed in first-order logic with identity, and requiring no set theory (1959):

the axioms should be read as universal closures; hence any free variables should be taken as tacitly universally quantified.

See examples:

Congruence axioms: Reflexivity of Congruence

$xy\equiv yx$,

and so on.