What is the formal definition of a topology on a non-empty set X?
Conditions
- X is a non-empty set.
- The collection consists of subsets of X.
- The three listed axioms must all hold.
Reasoning, step by step
- Identify the underlying set X.
- Define a collection of subsets of X.
- Verify that the empty set and X are members of the collection.
- Verify that arbitrary unions of members of the collection remain in the collection.
- Verify that finite intersections of members of the collection remain in the collection.
Example
For , the collection τ={∅,{a},{a,b},X} is a topology because it contains ∅ and X, unions of its members stay in τ, and finite intersections like {a}∩{a,b}={a} remain in τ.
Common misconceptions
- Assuming that any collection of subsets is a topology without checking the axioms.
- Confusing the requirement for arbitrary unions with the requirement for finite intersections.
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Related questions
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).
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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