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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}.

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).

A topology on a non-empty set X is a collection of subsets of X, called open subsets, that satisfies three specific axioms: (1) the empty set and the whole set X are in the collection; (2) the union of any family of sets in the collection is also in the collection; and (3) the intersection of any finite collection of sets in the collection is also in the collection.

Conditions: X is a non-empty set.; The collection consists of subsets of X.; The three listed axioms must all hold.

A topology on a non-empty set X is a collection of subsets of X, called open subsets, that satisfies three specific axioms: (1) the empty set and the whole set X are in the collection; (2) the union of any family of sets in the collection is also in the collection; and (3) the intersection of any finite collection of sets in the collection is also in the collection.

Conditions: X is a non-empty set.; The collection consists of subsets of X.; The three listed axioms must all hold.

In this context, a subset of X is open exactly when it is a member of the chosen topology τ. Openness is defined axiomatically by membership in τ, not by distance, intervals, or metric notions.

Conditions: A fixed set X and topology τ are chosen.; One is asking whether a subset of X belongs to τ.

In this context, a subset of X is open exactly when it is a member of the chosen topology τ. Openness is defined axiomatically by membership in τ, not by distance, intervals, or metric notions.

Conditions: A fixed set X and topology τ are chosen.; One is asking whether a subset of X belongs to τ.

To verify a collection τ is a topology on a finite set X, check three conditions: (1) confirm ∅ and X are in τ; (2) verify that the union of any subfamily of τ is in τ; and (3) verify that the intersection of any finite subfamily of τ is in τ.

Conditions: X is a finite set.; τ is a proposed collection of subsets of X.

To verify a collection τ is a topology on a finite set X, check three conditions: (1) confirm ∅ and X are in τ; (2) verify that the union of any subfamily of τ is in τ; and (3) verify that the intersection of any finite subfamily of τ is in τ.

Conditions: X is a finite set.; τ is a proposed collection of subsets of X.