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In the topology axiom, “finite intersection” means that the intersection of any finite number of open sets must also be an open set. It does not require infinite intersections to be open.

Conditions: Applying the third axiom of a topology.

In the topology axiom, “finite intersection” means that the intersection of any finite number of open sets must also be an open set. It does not require infinite intersections to be open.

Conditions: Applying the third axiom of a topology.

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}.