Topology Axioms — Symbolic Reference

A topology on a set is a collection satisfying:

The Three Axioms

. Contains ∅ and X

. Closed under arbitrary unions

Any collection of open sets (even uncountably many) — their union is open.

. Closed under finite intersections

Finitely many open sets — their intersection is open.

Prose vs Symbols (Munkres style)

Prose (Munkres)	Symbolic
"The empty set and X are in T"
"The union of the elements of any subcollection of T is in T"
"The intersection of the elements of any finite subcollection of T is in T"

Why finite for intersections?

Quick counterexample: In with standard topology,

Each interval is open, but is not open in .

That's why axiom requires finite intersections only.