The usual real topology contains far too many open sets to list, but intervals are enough to describe it. An arbitrary open set is a union of open intervals. Even intervals with rational endpoints suffice, because rational numbers can be chosen on both sides of any point within a given open interval.
A basis is such a stock of local building blocks. It does not need to contain every open set, and it need not be closed under taking unions.
Predict
Predict whether the intersection of two basis members must itself be a basis member. That stronger property is convenient but unnecessary. It is enough that every point in the intersection has a smaller basis member lying inside it.
Predict what happens if a proposed basis misses one point of X entirely. No union of its members can produce a neighborhood of that point, or even the whole space. Covering X is therefore a separate axiom.
Experiment
Compare two bases on the real line.
Basis shape
Neighborhood around x
Generated topology
(a,b)
An interval extending both sides
Usual
[a,b)
May begin exactly at x
Lower-limit
Singletons
The point alone
Discrete
For the lower-limit basis, identify why the endpoint x is an interior point of [x,x+1) in the generated topology even though it is not an interior point in the usual topology.
Understand
A family B is a basis if it covers X and satisfies local intersection refinement: whenever x belongs to B₁∩B₂, some B₃ contains x and lies in that intersection. Declare U open when it is a union of basis members.
The empty union gives the empty set, and covering gives X. Arbitrary unions of basis unions are still basis unions. For the intersection of two generated opens, take any point x, select a basis member from each open around x, and refine their intersection. The intersection is the union of all such refinements, so is open. Induction handles finite intersections. This proves the generated family is a topology.
The useful local test is therefore: U is open exactly when every x in U has some B in the basis with x∈B⊆U. One does not need to guess the complete union in advance.
To compare bases, the topology generated by B is finer than the topology generated by C precisely when every C member around every point contains a B member around that point. Necessity follows because C members are then B-open. Sufficiency follows because each C member becomes a union of B members. Keep the containment direction straight: smaller local building blocks allow more sets to be open.
Apply
Prove the lower-limit topology is strictly finer than the usual topology. Every usual interval is a union of half-open intervals contained in it, so all usual-open sets remain open. But [0,1) is lower-limit-open and not usual-open, since no ordinary neighborhood of zero fits inside it.
Repair “A basis must include the whole space.” It need only cover the space by its members. The collection of bounded real intervals is a basis even though no single member is the entire real line.
Continue
Bases turn continuity, density and countability questions into local refinement arguments. The next lesson starts with even fewer sets, a subbasis, and generates the required finite intersections automatically. Optional depth: prove rational-endpoint intervals form a countable usual basis, explicitly using rational density rather than just their visual resemblance to ordinary intervals.
Apply · explain · revise
Practice and reasoning
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On {1,2,3}, let B={{1},{2},{3}}. How many distinct unions of basis members are there?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A basis intersection condition says that if x belongs to B₁∩B₂:
Prove that unions of members of a covering family B satisfying local intersection refinement form a topology.
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Prove intervals with rational endpoints form a basis for the usual topology of R.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show [a,b), a<b, form a basis on R, and that this topology is strictly finer than the usual topology.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove B generates a topology finer than that generated by C exactly when each C-neighborhood of a point contains a B-neighborhood of that point.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
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