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A topology must satisfy all three axioms simultaneously. A collection can contain ∅ and X and be closed under unions, yet still fail to be a topology if it is not closed under finite intersections.

Conditions: The collection satisfies Axiom 1 (contains ∅ and X) and Axiom 2 (closed under unions).; The collection fails Axiom 3 (not closed under finite intersections).

A topology must satisfy all three axioms simultaneously. A collection can contain ∅ and X and be closed under unions, yet still fail to be a topology if it is not closed under finite intersections.

Conditions: The collection satisfies Axiom 1 (contains ∅ and X) and Axiom 2 (closed under unions).; The collection fails Axiom 3 (not closed under finite intersections).

This collection fails to be a topology because it is not closed under finite intersections. Specifically, the intersection of the two open sets {a, b} and {a, c} is {a}, but {a} is not included in the collection.

Conditions: $X = {a, b, c}$.; The proposed collection is τ = {∅, {a, b}, {a, c}, X}.

This collection fails to be a topology because it is not closed under finite intersections. Specifically, the intersection of the two open sets {a, b} and {a, c} is {a}, but {a} is not included in the collection.

Conditions: $X = {a, b, c}$.; The proposed collection is τ = {∅, {a, b}, {a, c}, X}.