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.
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.
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.