Why can a collection pass the union check but still fail to be a topology?
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).
Reasoning, step by step
- Check Axiom 1: Are ∅ and X in the collection?
- Check Axiom 2: Are unions of members in the collection?
- Check Axiom 3: Are finite intersections of members in the collection?
- If Axiom 3 fails, conclude the collection is not a topology, regardless of Axioms 1 and 2.
Example
In the second example, {a, b} ∪ {a, c} = X is in τ, so the union check passes, but {a, b} ∩ {a, c} = {a} is not in τ, so it fails to be a topology.
Common misconceptions
- Thinking that satisfying some axioms is enough to be a topology.
- Confusing the independence of the three axioms.
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Related questions
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.
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: .; The proposed collection is τ = {∅, {a, b}, {a, c}, X}.
A topological space is written as the ordered pair (X, τ) because the same underlying set X can carry different topologies. The topology τ specifies which subsets of X are considered open, so the structure of the space depends on both the set and the chosen collection of open subsets.
Conditions: X is a set.; τ is a topology on X.
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.
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